Let $f:\mathbb{R} \to \mathbb{R} $, and $f$ is differentiable in $x_0$.
Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of this neighborhood?
Which conditions say that this question is true?
2026-03-28 23:33:29.1774740809
Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of neighborhood?
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No. Standard example: Let $f(x)=x^2$ for rational $x$, $f(x)=0$ for irrational $x$. Then $f$ is differentiable at $0$ but it is not even continuous at any other point.