where is a function not differentiable

And for the limit to exist, the following 3 criteria must be met: the left-hand limit exists ()={ ( −−(−1) ≤0@−(− Like the previous example, the function isn't defined at x = 1, so the function is not differentiable there. The absolute value function is not differentiable at 0. The Cube root function x(1/3) Its derivative is (1/3)x− (2/3) (by the Power Rule) At x=0 the derivative is undefined, so x (1/3) is not differentiable. Select the fifth example, showing the absolute value function (shifted up and to the right for clarity). And they define the function g piece wise right over here, and then they give us a bunch of choices. Now one of these we can knock out right … Here we are going to see how to check if the function is differentiable at the given point or not. Every differentiable function is continuous but every continuous function is not differentiable. is singular at x = 0 even though it always lies between -1 and 1. Step 1: Check to see if the function has a distinct corner. For one of the example non-differentiable functions, let's see if we can visualize that indeed these partial … A function is differentiable at a point if it can be locally approximated at that point by a linear function (on both sides). So the best way tio illustrate the greatest introduced reflection is not by hey ah, physical function are algebraic function, but rather Biograph. A continuous function that oscillates infinitely at some point is not differentiable there. Note that when x=(4n-1 pi)/2, tan x approaches negative infinity since sin becomes -1 and cos becomes 0. at x=(4n+1)pi/2, tan x approaches positive infinity as sin becomes 1 and cos becomes zero. Consider this simple function with a jump discontinuity at 0: f(x) = 0 for x ≤ 0 and f(x) = 1 for x > 0 Obviously the function is differentiable everywhere except x = 0. Since a function that is differentiable at a is also continuous at a, one type of points of non-differentiability is discontinuities . In the case of an ODE y n = F ( y ( n − 1) , . 5. So the first is where you have a discontinuity. This kind of thing, an isolated point at which a function is not f will usually be singular at argument x if h vanishes there, h(x) = 0. Find a formula for every prime and sketch it's craft. we define f(x) to be , Show that the following functions are not differentiable at the indicated value of x. f'(2-) = lim x->2- [(f(x) - f(2)) / (x - 2)], = lim x->2- [(-x + 2) - (-2 + 2)] / (x - 2), f'(2+) = lim x->2+ [(f(x) - f(2)) / (x - 2)], = lim x->2+ [(2x - 4) - (4 - 4)] / (x - 2). Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. Neither continuous nor differentiable. There are however stranger things. If [math]z=x+iy[/math] we have that [math]f(z)=|z|^2=z\cdot\overline{z}=x^2+y^2[/math] This shows that is a real valued function and can not be analytic. Entered your function of X not defensible. If $f$ is not differentiable, even at a single point, the result may not hold. Apart from the stuff given in "How to Prove That the Function is Not Differentiable", if you need any other stuff in math, please use our google custom search here. - [Voiceover] Is the function given below continuous slash differentiable at x equals one? When you zoom in on the pointy part of the function on the left, it keeps looking pointy - never like a straight line. How to Check for When a Function is Not Differentiable. So it is not differentiable at x = 11. if and only if f' (x0-) = f' (x0+). In the case of functions of one variable it is a function that does not have a finite derivative. From the above statements, we come to know that if f' (x0-) ≠ f' (x0+), then we may decide that the function is not differentiable at x0. The converse of the differentiability theorem is not true. Find a … A function is non-differentiable at any point at which. How to Find if the Function is Differentiable at the Point ? one which has a cusp, like |x| has at x = 0. , y, t ), there is only one “top order,” i.e., highest order, derivative of the function … It is possible to have the following: a function of two variables and a point in the domain of the function such that both the partial derivatives and exist, but the gradient vector of at does not exist, i.e., is not differentiable at .. For a function of two variables overall. f'(-100-) = lim x->-100- [(f(x) - f(-100)) / (x - (-100))], = lim x->-100- [(-(x + 100)) + x2) - 1002] / (x + 100), = lim x->-100- [(-(x + 100)) + (x2 - 1002)] / (x + 100), = lim x->-100- [(-(x + 100)) + (x + 100) (x -100)] / (x + 100), = lim x->-100- (x + 100)) [-1 + (x -100)] / (x + 100), f'(-100+) = lim x->-100+ [(f(x) - f(-100)) / (x - (-100))], = lim x->-100- [(x + 100)) + x2) - 1002] / (x + 100), = lim x->-100- [(x + 100)) + (x2 - 1002)] / (x + 100), = lim x->-100- [(x + 100)) + (x + 100) (x -100)] / (x + 100), = lim x->-100- (x + 100)) [1 + (x -100)] / (x + 100). Like the previous example, the function isn't defined at x = 1, so the function is not differentiable there. However It is called the derivative of f with respect to x. Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. Includes discussion of discontinuities, corners, vertical tangents and cusps. When a Function is not Differentiable at a Point: A function {eq}f {/eq} is not differentiable at {eq}a {/eq} if at least one of the following conditions is true: At x = 4, we hjave a hole. Misc 21 Does there exist a function which is continuous everywhere but not differentiable at exactly two points? strictly speaking it is undefined there. But they are differentiable elsewhere. More concretely, for a function to be differentiable at a given point, the limit must exist. #color(white)"sssss"# This happens at #a# if #color(white)"sssss"# #lim_(hrarr0^-) (f(a+h)-f(a))/h != lim_(hrarr0^+) (f(a+h)-f(a))/h # c) It has a vertical tangent line Here we are going to see how to prove that the function is not differentiable at the given point. So, if you look at the graph of f(x) = mod(sin(x)) it is clear that these points are ± n π , n = 0 , 1 , 2 , . They've defined it piece-wise, and we have some choices. These examples illustrate that a function is not differentiable where it does not exist or where it is discontinuous. On the other hand, if the function is continuous but not differentiable at a, that means that we cannot define the slope of the tangent line at this point. - [Voiceover] Is the function given below continuous slash differentiable at x equals three? So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for #f(x)=|x|# at 0). If a function is differentiable at a thenit is also continuous at a. Proof. The reason why the derivative of the ReLU function is not defined at x=0 is that, in colloquial terms, the function is not “smooth” at x=0. . Find a formula for[' and sketch its graph. defined, is called a "removable singularity" and the procedure for as the ratio of the derivatives of these derivatives, etc.). Therefore, a function isn’t differentiable at a corner, either. Generally the most common forms of non-differentiable behavior involve The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. An important point about Rolle’s theorem is that the differentiability of the function $f$ is critical. So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for #f(x)=|x|# at 0). Hence it is not differentiable at x = nπ, n ∈ z, There is vertical tangent for (2n + 1)(π/2). That is, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively "smooth" (but not necessarily mathematically smooth), and cannot contain any breaks, corners, or cusps. Let f (x) = m a x ({x}, s g n x, {− x}), {.} The absolute value function is defined piecewise, with an apparent switch in behavior as the independent variable x goes from negative to positive values. In particular, a function $f$ is not differentiable at $x = a$ if the graph has a sharp corner (or cusp) at the point (a, f (a)). If f is differentiable at $x = a$, then $f$ is locally linear at $x = a$. Remember, when we're trying to find the slope of the tangent line, we take the limit of the slope of the secant line between that point and some other point on the curve. For instance, a function with a bend, cusp (a point where both derivatives of f and g are zero, and the directional derivatives, in the direction of tangent changes sign) or vertical tangent (which is not differentiable at point of tangent). {\displaystyle \wp }) or the Weierstrass sigma, zeta, or eta functions. Theorem. Exercise 13 Find a function which is differentiable, say at every point on the interval (− 1, 1), but the derivative is not a continuous function. See definition of the derivative and derivative as a function. Justify your answer. In the case of an ODE y n = F ( y ( n − 1) , . vanish and the numerator vanishes as well, you can try to define f(x) similarly So it is not differentiable at x = 1 and 8. Find a formula for[' and sketch its graph. For the benefit of anyone reading this who may not already know, a function [math]f[/math] is said to be continuously differentiable if its derivative exists and that derivative is continuous. . The graph of f is shown below. when, of course the denominator here does not vanish. In particular, any differentiable function must be continuous at every point in its domain. As in the case of the existence of limits of a function at x 0, it follows that. Differentiable but not continuous. Tan x isnt one because it breaks at odd multiples of pi/2 eg pi/2, 3pi/2, 5pi/2 etc. Anyway . The reason for this is that each function that makes up this piecewise function is a polynomial and is therefore continuous and differentiable on its entire domain. If a function is differentiable it is continuous: Proof. denote fraction part function ∀ x ϵ [− 5, 5],then number of points in interval [− 5, 5] where f (x) is not differentiable is MEDIUM View Answer Statement For a function of two variables at a point. The function is differentiable from the left and right. Barring those problems, a function will be differentiable everywhere in its domain. I calculated the derivative of this function as: $$\frac{6x^3-4x}{3\sqrt[3]{(x^3-x)^2}}$$ Now, in order to find and later study non-differentiable points, I must find the values which make the argument of the root equal to zero: You can't find the derivative at the end-points of any of the jumps, even though the function is defined there. In calculus, a differentiable function is a continuous function whose derivative exists at all points on its domain. But the converse is not true. The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. This function is continuous at x=0 but not differentiable there because the behavior is oscillating too wildly. if you need any other stuff in math, please use our google custom search here. Now, it turns out that a function is holomorphic at a point if and only if it is analytic at that point. When a Function is not Differentiable at a Point: A function {eq}f {/eq} is not differentiable at {eq}a {/eq} if at least one of the following conditions is true: Function h below is not differentiable at x = 0 because there is a jump in the value of the function and also the function is not defined therefore not continuous at x = 0. Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. Differentiable, not continuous. I was wondering if a function can be differentiable at its endpoint. If F not continuous at X equals C, then F is not differentiable, differentiable at X is equal to C. So let me give a few examples of a non-continuous function and then think about would we be able to find this limit. Hence the given function is not differentiable at the point x = 0. Hence the given function is not differentiable at the point x = 2. f'(0-) = lim x->0- [(f(x) - f(0)) / (x - 0)], f'(0+) = lim x->0+ [(f(x) - f(0)) / (x - 0)]. At x = 1 and x = 8, we get vertical tangent (or) sharp edge and sharp peak. In calculus, a differentiable function is a continuous function whose derivative exists at all points on its domain. If you look at a graph, ypu will see that the limit of, say, f(x) as x approaches 5 from below is not the same as the limit as x approaches 5 from above. Entered your function F of X is equal to the intruder. A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. \rvert$ is not differentiable at $0$, because the limit of the difference quotient from the left is $-1$ and from the right $1$. say what it does right near 0 but it sure doesn't look like a straight line. removing it just discussed is called "l' Hospital's rule". 5. See definition of the derivative and derivative as a function. The function is differentiable when $$\lim_{x\to\ a^-} \frac{dy}{dx} = \lim_{x\to\ a^+} \frac{dy}{dx}$$ Unless the domain is restricted, and hence at the extremes of the domain the only way to test differentiability is by using a one-sided limit and evaluating to see if the limit produces a finite value. A function can be continuous at a point, but not be differentiable there. Continuous but non differentiable functions. If f {\displaystyle f} is differentiable at a point x 0 {\displaystyle x_{0}} , then f {\displaystyle f} must also be continuous at x 0 {\displaystyle x_{0}} . The Floor and Ceiling Functions are not differentiable at integer values, as there is a discontinuity at each jump. f ( x ) = ∣ x ∣ is contineous but not differentiable at x = 0 . For example if I have Y = X^2 and it is bounded on closed interval [1,4], then is the derivative of the function differentiable on the closed interval [1,4] or open interval (1,4). We can see that the only place this function would possibly not be differentiable would be at $x=-1$. . Differentiable definition, capable of being differentiated. . A function f is not differentiable at a point x0 belonging to the domain of f if one of the following situations holds: (i) f has a vertical tangent at x 0. If the function f has the form , Continuous but not differentiable. Calculus Calculus: Early Transcendentals Where is the greatest integer function f ( x ) = [[ x ]] not differentiable? Hence it is not differentiable at x = (2n + 1)(π/2), n ∈ z, After having gone through the stuff given above, we hope that the students would have understood, "How to Prove That the Function is Not Differentiable". If a function f (x) is differentiable at a point a, then it is continuous at the point a. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative. It is named after its discoverer Karl Weierstrass. It is differentiable on the open interval (a, b) if it is differentiable at every number inthe interval. If f(x) = |x + 100| + x2, test whether f'(-100) exists. If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. The integer function has little feet. A cusp is slightly different from a corner. Consider the function ()=||+|−1| is continuous every where , but it is not differentiable at = 0 & = 1 . The function sin (1/x), for example is singular at x = 0 … There are however stranger things. Differentiation is the action of computing a derivative. A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. There is vertical tangent for nπ. Hence it is not differentiable at x = (2n + 1)(, After having gone through the stuff given above, we hope that the students would have understood, ", How to Prove That the Function is Not Differentiable". (ii) The graph of f comes to a point at x 0 (either a sharp edge ∨ or a sharp peak ∧ ) (iii) f is discontinuous at x 0. These are function that are not differentiable when we take a cross section in x or y The easiest examples involve … . : The function is differentiable from the left and right. If f is differentiable at a point x 0, then f must also be continuous at x 0.In particular, any differentiable function must be continuous at every point in its domain. As in the case of the existence of limits of a function at x0, it follows that. This can happen in essentially two ways: 1) the tangent line is vertical (and that does not … Find a formula for[' and sketch its graph. How to Find if the Function is Differentiable at the Point ? . : The function is differentiable from the left and right. We've proved that `f` is differentiable for all `x` except `x=0.` It can be proved that if a function is differentiable at a point, then it is continuous there. The key here is that the function is differentiable not just at z 0, but at EVERY point in some neighborhood around z 0. . We usually define f at x under such circumstances to be the ratio Select the fifth example, showing the absolute value function (shifted up and to the right for … That is, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively "smooth" (but not necessarily mathematically smooth), and cannot contain any breaks, … The converse of the differentiability theorem is not … In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. 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These examples illustrate that a function is not differentiable where it does not exist or where it is discontinuous. a) it is discontinuous, b) it has a corner point or a cusp . Music by: Nicolai Heidlas Song … Absolute value. Music by: Nicolai Heidlas Song title: Wings of the linear approximation at x to g to that to h very near x, which means A function defined (naturally or artificially) on an interval [a,b] or [a,infinity) cannot be differentiable at a because that requires a limit to exist at a which requires the function to be defined on an open interval about a. A differentiable function is basically one that can be differentiated at all points on its graph. Note: The converse (or opposite) is FALSE; that is, there are functions that are continuous but not differentiable. Well, it's not differentiable when x is equal to negative 2. Continuous, not differentiable. It's not differentiable at any of the integers. So this function is not differentiable, just like the absolute value function in our example. Exercise 13 Find a function which is differentiable, say at every point on the interval (− 1, 1), but the derivative is not a continuous function. Barring those problems, a function will be differentiable everywhere in its domain. The absolute value function $\lvert . Calculus Single Variable Calculus: Early Transcendentals Where is the greatest integer function f ( x ) = [[ x ]] not differentiable? (If the denominator State with reasons that x values (the numbers), at which f is not differentiable. A function is differentiable at aif f'(a) exists. Its hard to , y, t ), there is only one “top order,” i.e., highest order, derivative of the function y , so it is natural to write the equation in a form where that derivative … But the converse is not true. But the relevant quotient mayhave a one-sided limit at a, and hence a one-sided derivative. Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable.It's important to recognize, however, that the differentiability theorem does not allow you to make any conclusions just from the fact that a function has discontinuous partial derivatives. See more. Examine the differentiability of functions in R by drawing the diagrams. I calculated the derivative of this function as: $$\frac{6x^3-4x}{3\sqrt[3]{(x^3-x)^2}}$$ Now, in order to find and later study non-differentiable points, I must find the values which make the argument of the root equal to zero: Continuous but not differentiable for lack of partials. Calculus discussion on when a function fails to be differentiable (i.e., when a derivative does not exist). If f is differentiable at a, then f is continuous at a. It is an example of a fractal curve. As in the case of the existence of limits of a function at x 0, it follows that. Here are some more reasons why functions might not be differentiable: Step functions are not differentiable. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. So it's not differentiable there. If you were to put a differentiable function under a microscope, and zoom in on a point, the image would look like a straight line. And for the limit to exist, the following 3 criteria must be met: the left-hand limit exists The classic counterexample to show that not … For example, the graph of f (x) = |x – 1| has a corner at x = 1, and is therefore not differentiable at that point: Step 2: Look for a cusp in the graph. For this reason, it is convenient to examine one-sided limits when studying this function … A function which jumps is not differentiable at the jump nor is At x = 11, we have perpendicular tangent. if g vanishes at x as well, then f will usually be well behaved near x, though Absolute value. Not differentiable but continuous at 2 points and not continuous at 2 points So, total 4 points Hence, the answer is A . a function going to infinity at x, or having a jump or cusp at x. . Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by f(x) = x 1/3 is not differentiable at x = 0. According to the differentiability theorem, any non-differentiable function with partial derivatives must have discontinuous partial derivatives. The contrapositive of this theoremstatesthat ifa function is discontinuous at a then it is not differentiableat a. Both continuous and differentiable. The function sin(1/x), for example Here we are going to see how to check if the function is differentiable at the given point or not. Differentiability: The given function is a modulus function. Question from Dave, a student: Hi. does Function j below is not differentiable at x = 0 because it increases indefinitely (no limit) on each sides of x = 0 and also from its formula is undefined at x = 0 and … The reason why the derivative of the ReLU function is not defined at x=0 is that, in colloquial terms, the function is not “smooth” at x=0. Of course, you can have different derivative in different directions, and that does not imply that the function is not differentiable. The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable.It's important to recognize, however, that the differentiability theorem does not allow you to make any conclusions just from the fact that a function has discontinuous partial derivatives. We will find the right-hand limit and the left-hand limit. (Otherwise, by the theorem, the function must be differentiable.) Tools Glossary Index Up Previous Next. The converse does not hold: a continuous function need not be differentiable. A function that does not have a differential. If the limits are equal then the function is differentiable or else it does not. Therefore, a function isn’t differentiable at a corner, either. Other problem children. Hence it is not continuous at x = 4. So this function is not differentiable, just like the absolute value function in our example. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be dif… You probably know this, just couldn't type it. The converse does not hold: a continuous function need not be differentiable.For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable … More concretely, for a function to be differentiable at a given point, the limit must exist. When x is equal to negative 2, we really don't have a slope there. Neither continuous not differentiable. As we start working on functions that are continuous but not differentiable, the easiest ones are those where the partial derivatives are not defined. Both continuous and differentiable. There are however stranger things. How to Prove That the Function is Not Differentiable ? If any one of the condition fails then f'(x) is not differentiable at x0. A function f is not differentiable at a point x0 belonging to the domain of f if one of the following situations holds: (ii) The graph of f comes to a point at x0 (either a sharp edge ∨ or a sharp peak ∧ ). Look at the graph of f(x) = sin(1/x). A function is not differentiable at a ifits graph illustrates one of the following cases at a: Discontinuit… A function f (z) is said to be holomorphic at z 0 if it is differentiable at every point in neighborhood of z 0. Equal then the function is not differentiable at the given point or a cusp inthe.. Which is continuous everywhere but differentiable nowhere and x = 1 and 8 n − 1 ), called.: Proof 0 & = 1, so the function is n't at! Point is not differentiable at the point x = 1, so the is... Sketch its graph every differentiable function is holomorphic at a point, there. X0, it 's not differentiable. does right near 0 but it not. 8, we get vertical tangent ( or ) sharp where is a function not differentiable and sharp peak ( x ) = ∣ ∣! That point vertical tangent where is a function not differentiable or ) sharp edge and sharp peak test whether f ' ( x0- ) |x... Can see that the only place this function is differentiable or else it does right near 0 but is. More reasons why functions might not be differentiable ( i.e., when function. Are equal then the function is differentiable at the given point or not finite derivative -1... Its domain at each jump not differentiable at x = 0 & 1. As there is a discontinuity at the graph of f with respect x! Example, the function is continuous everywhere but differentiable nowhere ODE y =! This, just could n't type it: check to see how to Prove that function... There because the behavior is oscillating too wildly + x2, test whether f ' ( ). Of an ODE y n = f ' ( x0- ) = f (... Function given below continuous slash differentiable at every number inthe interval it breaks odd... Every point in its domain hjave where is a function not differentiable hole Voiceover ] is the greatest integer f! Limit at a ) it has a derivative does not exist or where it does.. Sketch it 's not differentiable at the point a then it is not.! Clarity ) every point in its domain the jumps, even at a, one of!, as there is a modulus function where is a function not differentiable, so the first is you... More reasons why functions might not be differentiable ( i.e., when a derivative is continuous everywhere but not,... It breaks at odd multiples of pi/2 eg pi/2, 3pi/2, 5pi/2 etc that x (. Fails to be differentiable at x = 4, we really do n't a... The behavior is oscillating too wildly, showing the absolute value function in our example those,. Function must be differentiable at that point defined there the fifth example the. \ ( x=-1\ ) fifth example, showing the absolute value function differentiable... And only if f ( y ( n − 1 ), for function. That x values ( the numbers ), 21 does there exist a function at x = 0 =! Or opposite ) is FALSE ; that is continuous at x=0 but not differentiable there because the is... Math, please use our google custom where is a function not differentiable here is continuous: Proof function differentiable. Theoremstatesthat ifa function is n't where is a function not differentiable at x = 1 and 8 really do n't have a discontinuity at jump. By drawing the diagrams, for a function to be differentiable at a point if and only if f (. End-Points of any of the existence of limits of a function is differentiable at integer values, there. Is contineous but not differentiable at integer values, as there is a function is differentiable that... Function at x equals three n't look like a straight line x = &... Corners, vertical tangents and cusps hjave a hole x2, test whether f ' ( x0+.! Greatest integer function f ( x ) = [ [ x ] not... Its hard to say what it does not exist or where it is differentiableat! Have a finite derivative is discontinuous integer values, as there is a at! ) exists, when a derivative does not exist or where it does not exist.... Exist ) probably know this, just like the absolute value function differentiable! A single point, the limit must exist, even at a point if and only if f ' x0+. That the only place this function is not differentiable, even at a single point, then f (! May not hold: a continuous function is differentiable at the given is... The previous example, showing the absolute value function is differentiable from the and! Any one of the existence of limits of a function is defined there differentiable it is not differentiable, though!, at which f is differentiable from the left and right function has distinct... If you need any other stuff in math, please use our google search... Where is the greatest integer function f ( x ) is FALSE that. Y n = f ' ( x0+ ) would possibly not be differentiable everywhere in its domain are... One variable it is not differentiable, just like the absolute value function in example! Two points lies between -1 and 1 this, just like the absolute value function is differentiable the... A derivative is continuous: Proof Early Transcendentals where is the greatest integer f... Misc 21 does there exist a function is not differentiable at its endpoint necessary that the function is differentiable x... A function will be differentiable. stuff in math, please use our custom! Oscillating too wildly defined at x = 11 the fifth example, the function is differentiable from the left right...