The title is self explanatory, does continuity of $f$ at a point require the existence of $f$ in a neighbourhood of that point?
2026-02-23 08:42:18.1771836138
Does continuity of $f$ at a point require the existence of $f$ in a neighbourhood of that point?
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Usually, the only thing that it is required about the point $p$ in order to say that a function $f\colon D\longrightarrow\Bbb R$ is continuous at $p$ is is an accumulation point of $D$ which belongs to $D$. In other words, $p\in D$ and every neighbourhood contains some point of $D$ other than $p$. So, in particular, no, $f$ doesn't have to be defined on a neighbourhood of $p$.