In fact, it is differentiable at every point other than on the Cantor set, which is a set of measure zero. 28 comments.

Naively, I would've expected it to be differentiable on the complement of the Cantor set again, just like the original staircase. If any one of the condition fails then f'(x) is not differentiable at x 0.
In Chapter 4 space filling curves are define. Thus, we have Proposition 2.11. There exists a differentiable function f with an uniformly bounded derivative f ′ such that G and f are Lebesgue equivalent.

The Cantor function is a function that is continuous, differentiable, increasing, non-constant, and the derivative is zero everywhere except at a set with length zero.

Moreover, if f is differentiable at every point of a set which has positive measure in each interval from [a,b], then f is a sum of two superpositions, f =f1 f2 +f3 f4, where fi,i=1,...,4, are absolutely continuous [2].
Volterra's function is differentiable everywhere just as f (as defined above) is. Ash, J. Marshall, Real Analysis Exchange, 2005 So the fat Cantor staircase must be differentiable on some points in the Cantor set!

At first, some examples of continuous nowhere differentiable functions are discussed.

Recall the definition of the Cantor set: Let = [ 1/3, 2/3] be the middle third of the interval [0, 1]. In addition, this result cannot be improved to countable: see Cantor function .

Volterra's function is differentiable everywhere just as f (as defined above) is.

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But they are differentiable elsewhere.

Next, space-filling functions, which are …

They have no gaps or pointy bits.

This Demonstration runs eight iterations of the Cantor function. This thread is archived. Consider the two functions ϕ1, ϕ2 pictured in Figure 1.2. But this contradicts the theorem that says monotone functions are differentiable almost everywhere.

28 comments.

This is an interesting example of how identifying a random variable with its PDF can lead us astray. This is because any weak derivative of c would have to be equal almost everywhere to the classical derivative of c, which is zero almost everywhere. exist and f' (x 0-) = f' (x 0 +) Hence if and only if f' (x 0-) = f' (x 0 +). We now recall a notion of set of varying monotonicity [30, Definition 3.7]. Thus, we have Proposition 2.11.

A remark on continuous, nowhere differentiable functions Okamoto, Hisashi, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2005; Least Squares and Approximate Differentiation Gordon, Russell A., Real Analysis Exchange, 2012; Differential geometry of generalized lagrangian functions Okubo, Katsumi, Journal of Mathematics of Kyoto University, 1991

the set of numbers in such that is not differentiable in has Lebesgue measure zero. You can zoom in close to the origin to see the fractal nature of the function. As in the case of the existence of limits of a function at x 0, it follows that. The Cantor function helps us understand what "nice enough" means. I know that the Cantor function is continuous everywhere and has zero derivative almost everywhere, but it's non-differentiable at uncountably many points.

The main purpose of this note is to verify that the Hausdorff dimension of the set of points N * at which the Cantor function is not differentiable is [ln(2)/ln(3)] 2 . 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. The function is differentiable from the left and right.

In fact, it is differentiable at every point other than on the Cantor set, which is a set of measure zero. Some examples of such curves, construction and proof of the fact that they are continuous but nowhere differentiable are discussed.

Is there any (non-constant) function that's differentiable everywhere and has zero derivative almost everywhere? save hide report.

Then Cantor function is defined and it is proved that the Cantor function is nowhere differentiable on Cantor set.

64% Upvoted. On the Cantor set the function is not differentiable and so has no PDF. Moreover, if f is differentiable at every point of a set which has positive measure in each interval from [a,b], then f is a sum of two superpositions, f =f1 f2 +f3 f4, where fi,i=1,...,4, are absolutely continuous [2]. Secondly, the Lebesgue-Cantor singular function is considered, which is continuous but the fundamental theorem of calculus is not valid for this function.