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 $B$.
Continuous Function
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real-analysis
general-topology
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1What's the definition of continuity you are using? – 2012-04-28
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2Isn'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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1$f$ is actually bounded on every open ball (because it is bounded on every closed ball). – 2012-04-28
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5He only stated continuity at a single point, so it is not necessarily bounded on any open ball. @MichaelGreinecker – 2012-04-28
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0It'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. @DavidWallace – 2012-04-28
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0Sure. Read my "more-or-less" as "trivially consequent upon". – 2012-04-28