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For a function $f: U \to \mathbb{R}$ where $U$ is a subset of $\mathbb{R}$, it seems like that it being continuous at a point doesn't imply that there is a neighbourhood of the point where it can be continuous. Similarly, it seems like that it being differentiable at a point doesn't imply that there is a neighbourhood of the point where it can be differentiable. I was wondering if there are some counterexamples to confirm the above?

Added:

What are some necessary and/or sufficient conditions for continuity/differentiability at a point and in some neighbourhood of the point to be equivalent?

Can the case of continuity be generalized to mappings between topological spaces?

Thanks and regards!

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    What is $U$? A topological space, or just a metric space?2012-02-10
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    $U$ is a subset of $\mathbb{R}$2012-02-10
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    Related: http://math.stackexchange.com/q/228392012-02-11
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    Hmm, I hadn't notice that the linked question was asked by the same person. Tim, did you realize you had already asked this question?2012-02-12
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    @JonasMeyer: I didn't. I didn't see your last comment link to my old question either. But I do now.2012-02-12

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