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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?

Nate
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    One famous example is the Weierstrass function, which is continuous everywhere but differentiable nowhere. – Minus One-Twelfth Feb 16 '19 at 01:18
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    van der Waerden also constructed a nowhere differentiable but continuous function, which seems to be a bit easier to understand. The construction could be seen here. – xbh Feb 16 '19 at 02:36
  • The sample paths of Brownian motion are continuous everywhere but differentiable nowhere. – Math1000 Feb 16 '19 at 05:17
  • For some results involving pathological differentiability behavior (much more refined than simply "not differentiable") of most continuous functions (in the sense of Baire category, and Wiener measure, and quasi everywhere, and Haar null, and $\sigma$-porous, and HP-small), see the second half of my answer to Generic Elements of a Set. – Dave L. Renfro Feb 16 '19 at 09:05

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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.

Markus Scheuer
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