Pathological, continuous functions

Question

Today in my introduction to measure theory course, the professor said that often when we think of continuity, what we're actually thinking about is smooth functions. We've studied the Cantor set and its variations, and he said we ought to think of continuous functions like the Cantor-Lebegsue function more often when we think continuity.

I was wondering what are other example of "pathological" yet continuous functions? Functions that really help enforce the idea of continuity as distinct from smoothness or even just differentiable?

Answer 1

The following theorem is stated in chapter 8 Blumberg's Theorem and Sierpiński-Zygmund function in Strange Functions in Real Analysis by A.B. Kharazishvili. It gives a connection between arbitrary functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ and continuous restrictions $f|D$ on some sets $D\subseteq\mathbb{R}$ which are not small (in a certain sense).

Theorem (Blumberg) Let $f$ be an arbitrary function acting from $\mathbb{R}$ into $\mathbb{R}$. Then there exists an everywhere dense subset $X$ of $\mathbb{R}$ such that the function $f|X$ is continuous.