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?
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