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.

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.

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TopNewestThe 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.

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Yup, that's what I was thinking of.

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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.

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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.

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Yes yes I was trying to point that out only.

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Oh sorry I didn't see your post is already replied.

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Weierstrass function is such a function

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