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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
1 year, 1 month 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. Brian Charlesworth · 1 year, 1 month ago

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@Brian Charlesworth Yup, that's what I was thinking of. Hobart Pao · 1 year, 1 month ago

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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. Kuldeep Guha Mazumder · 1 year ago

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@Kuldeep Guha Mazumder 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. Brian Charlesworth · 1 year ago

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@Brian Charlesworth Yes yes I was trying to point that out only. Kuldeep Guha Mazumder · 1 year ago

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Oh sorry I didn't see your post is already replied. Ravi Dwivedi · 1 year, 1 month ago

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Weierstrass function is such a function Ravi Dwivedi · 1 year, 1 month ago

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