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Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is continuous at $P$. Can anyone help me prove that there is an open ball $B$ in $\mathbb{R}^n$ with center $P$ such that $f$ is bounded on $B$.

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    What's the definition of continuity you are using?2012-04-28
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    Isn't that more-or-less exactly the definition of continuous? Let $\epsilon$ be whatever you like, find $\delta$ and let $B$ have radius $\delta$. Or have I missed something subtle?2012-04-28
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    $f$ is actually bounded on every open ball (because it is bounded on every closed ball).2012-04-28
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    He only stated continuity at a single point, so it is not necessarily bounded on any open ball. @MichaelGreinecker2012-04-28
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    It's also clearly not "exactly the definition of continuity," since it is easy to find functions that satisfy this but are not continuous. It is, however, a direct consequence of the definition of continuity. @DavidWallace2012-04-28
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    Sure. Read my "more-or-less" as "trivially consequent upon".2012-04-28

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