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?
Answer
A prominent example of such a function is the Weierstrass function. In general examining the difference quotients is a very good start for showing non-differentiability in such severe cases.
There is a well-written report on the Weierstrass function and continuous everywhere, differentiable nowhere functions in general where also a proof of these properties is presented for the Weierstrass function in particular.
From a quick reading it seems that they also examine the difference quotient to infer a contradiction to the basic definition of differentiability.
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.
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