Let $F$ be a strictly increasing function on $S$, a subset of the real line. If you know that $F(S)$ is closed, prove that $F$ is continuous.
If $F$ is strictly increasing with closed image, then $F$ is continuous
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real-analysis
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3Intuitively, since $F(S)$ is closed and $F$ is increasing, there are no jumps: if there's a jump from $a$ to $b$ then one of the endpoints would be a limit point of $F(S)$ not in $F(S)$; since $F$ is increasing, $F$ will never equal $a$ again. – 2010-11-05
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0@Yuval: Well put. That is what I had in mind with my first proof, but you have better conveyed the intuition. – 2010-11-05