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Let $f:\mathbb{R^n}\to\mathbb{R^m}$ be a function such that the image of any closed bounded set is closed and bounded. Must $f$ be continuous?

2 Answers 2

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No.

For example take $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by

$f(x) = 0$ if $x < 0$

$f(x) = 1$ if $x \geq 0$

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    Wow, this is a case o$f$ two solutions independently and simultaneously being developed i$f$ I ever saw one :) I'll delete mine since yours is timestamped earlier.2012-11-01
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Assume that the image by $f$ of any set is closed and bounded, for example because $f(\mathbb R^n)=\{a,b\}$ for some $a\ne b$ in $\mathbb R^m$. Such functions $f$ need not be continuous (example?).

(Once you will have exhibited such a function discontinuous at a point, say, you might try to find one which is discontinuous everywhere, since the idea is the same.)

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    If $n=m=1$, try $f(x)=\sin(1/x)$ for $x\ne0$ and $f(0)=0$.2012-11-01