The relationship between differentiability and continuity

Name a function that is everywhere continuous but nowhere differentiable. Or, if such a function does not exist, explain why.

Note by Hobart Pao
2 years, 7 months ago

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The Weierstrass function is the most commonly cited example. Here and here are other examples, not to mention fractals.

It could be possible that "almost all" everywhere continuous functions are nowhere differentiable, but I'll have to give that one a bit more thought.

- 2 years, 7 months ago

Yup, that's what I was thinking of.

- 2 years, 7 months ago

If you, by the word 'everywhere', mean everywhere in its domain, then I can give you a much simpler function: Define the function $$f:N\rightarrow R$$ as $$f\left( n \right)=n,\forall n\in N$$. Clearly, since each point in its domain is isolated, we can say that it is continuous 'everywhere' in its domain by default, but not differentiable anywhere.

- 2 years, 5 months ago

Good point. By most definitions of continuity a discrete function, (i.e., a function in which its domain is at most countable), it is considered "vacuously true" that the function is continuous at all of its (isolated) points. We could also then consider the function $$f:\mathbb{Q} \rightarrow \mathbb{R}, f(x) = x$$ for all $$x \in \mathbb{Q}$$ as an example.

- 2 years, 5 months ago

Yes yes I was trying to point that out only.

- 2 years, 5 months ago

- 2 years, 7 months ago

Weierstrass function is such a function

- 2 years, 7 months ago