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$f$ is a real valued function. $f$ is differentiable at a point say $a$.

Does that imply $f$ is continuous in a neighbourhood of $a$?

I think the answer is false, but I can not find a counterexample.

By definition, if a function is differentiable at a point a then the function is defined in a neighbourhood of $a$. Am I right?

We can only conclude that $f$ is continuous at that point $a$.

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Think of $$x\mapsto x^2\mathbb{1}_{\Bbb Q}(x)$$ It's only continuous at $0$, but the $x^2$ presses the curve to still have a zero derivative at $0$.

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