Continuous everywhere differentiable nowhere function

If I want to prove that a function is continuous everywhere differentiable nowhere, then I prove that the axioms for continuity hold, proving that the limit values in every point is equal to the evaluated function (for example if f is never undefined and limit is the same as f(x) in every point then it is continuous).

Then to prove that it is never differentiable, should I use the definition of the derivate and prove that the derivate never exists? How is that usually done?

• Your profile picture looks like a swastika Jul 26, 2018 at 5:19
• In order to prove that a function is never differentiable you must show that the derivative does not exist at any point, and yes, you should do it by the definition of the derivative. Jul 26, 2018 at 7:26
• You may be interested in the Weierstrass Function. Jul 26, 2018 at 9:14

As of the property of being continuous everywhere, the procedure depends pretty much on the function your examining and what kind of analytic representation. As the Weierstrass function in particular is defined as a functional series, they use well-known correspondences between uniform convergence of functional sequences/series and continuity as well as the so called Weierstrass $M$-test (a test to determine whether a series of functions converges uniformly) to derive the respective results.