Discontinuous derivative.

Question

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.

Answer 1

I guess that you are looking for a continuous function $ f: \mathbb{R} \to \mathbb{R} $ such that $ f $ is differentiable everywhere but $ f' $ is ‘as discontinuous as possible’.

We have the following theorem in real analysis.

Theorem 1 If $ f: \mathbb{R} \to \mathbb{R} $ is differentiable everywhere, then the set of points in $ \mathbb{R} $ where $ f' $ is continuous is non-empty. More precisely, the set of all such points is a dense $ G_{\delta} $-subset of $ \mathbb{R} $.

Note: A $ G_{\delta} $-subset of $ \mathbb{R} $ is just the intersection of a countable collection of open subsets of $ \mathbb{R} $.

The proof of Theorem 1 is an application of the Baire Category Theorem, and it can be found in Munkres’ Topology or Measure and Category by Oxtoby. By this theorem, it is therefore impossible to find an $ f: \mathbb{R} \to \mathbb{R} $ whose derivative exists but is discontinuous everywhere.

There is another theorem that provides a necessary and sufficient condition for a set $E$ to be the set of discontinuities of some derivative.

Theorem 2 A set $E$ is the set of discontinuities of some derivative if and only if $E$ is a meagre $ F_{\sigma} $-subset of $ \mathbb{R} $.

Note: An $ F_{\sigma} $-subset of $ \mathbb{R} $ is just the union of a countable collection of closed subsets of $ \mathbb{R} $.

Let me end off with a non-trivial example to add to yours. Volterra’s Function is differentiable everywhere, but its derivative is discontinuous on a set of positive measure, not just at a single point.