Ash, J. Marshall, Real Analysis Exchange, 2005 ; This thread is archived. The Cantor function. Secondly, the Lebesgue-Cantor singular function is considered, which is continuous but the fundamental theorem of calculus is not valid for this function. The Cantor function helps us understand what "nice enough" means. So, the derivative of \(f\) is \(f'(x) = 3x^2 + 6x + 2\). Some examples of such curves, construction and proof of the fact that they are continuous but nowhere differentiable are discussed. So the fat Cantor staircase must be differentiable on some points in the Cantor set! Thus, we have Proposition 2.11. Volterra's function is differentiable everywhere just as f (as defined above) is. I know that the Cantor function is continuous everywhere and has zero derivative almost everywhere, but it's non-differentiable at uncountably many points. In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. 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. The Cantor function c does not have a weak derivative, despite being differentiable almost everywhere. 2 In this example we showed that this function does not contain zero. the set of numbers in such that is not differentiable in has Lebesgue measure zero. 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 rules of differentiation tell us that the derivative of \(x^3\) is \(3x^2\), the derivative of \(x^2\) is \(2x\), and the derivative of \(x\) is \(1\). share. Is there any (non-constant) function that's differentiable everywhere and has zero derivative almost everywhere? Is there any (non-constant) function that's differentiable everywhere and has zero derivative almost everywhere? save hide report. This is the most difficult function in our repertoire and can be found, for example, in Kolmogorov and Fomin. 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]. Is the function \(f(x) = x^3 + 3x^2 + 2x\) differentiable?

Volterra's function is differentiable everywhere just as f (as defined above) is. In fact, it is differentiable at every point other than on the Cantor set, which is a set of measure zero. 64% Upvoted. 1.9 The Cantor–Lebesgue Function We will construct an important function in this section through an iterative procedure that is related to the construction of the Cantor set as given in Example 1.8. It is either identically zero or not defined. In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous.It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. On the Cantor set the function is not differentiable and so has no PDF. What we see is that, for a Cantor random variable, we cannot make any sensible definition for the PDF.

exist and f' (x 0-) = f' (x 0 +) Hence if and only if f' (x 0-) = f' (x 0 +). Recall the definition of the Cantor set: Let = [ 1/3, 2/3] be the middle third of the interval [0, 1]. Where is this fat Cantor staircase differentiable? This is an interesting example of how identifying a random variable with its PDF can lead us astray. share. On the critical case of Okamoto’s continuous non-differentiable functions Kobayashi, Kenta, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2009; An Lp differentiable non-differentiable function. 64% Upvoted. Consider the two functions ϕ1, ϕ2 pictured in Figure 1.2. The Floor and Ceiling Functions are not differentiable at integer values, as there is a discontinuity at each jump.

This thread is archived.

The function is differentiable from the left and right. 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 Cantor function, also known as the Cantor Staircase, is a bizarre function that is continuous and has a derivative of 0 at every point where it is differentiable. Example. Differentiable functions are nice, smooth curvy animals. Thus, we have Proposition 2.11. You can zoom in close to the origin to see the fractal nature of the function. if is a monotonic function defined on an interval, then is differentiable almost everywhere on , i.e. Naively, I would've expected it to be differentiable on the complement of the Cantor set again, just like the original staircase.