Let $f:I\rightarrow R$ be a monotonically increasing function on an open interval.If the image of this interval is an interval then would $f$ be continuous? For the case when $f(I)$ is open then I can deduct continuity of $f$.But What if $f(I)$ closed?
Continuity of a monotonically increasing function
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analysis
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1When you say *monotonically increasing*, do you mean that $x
implies that $f(x) ($f$ is strictly increasing), or do you mean that $x\le y$ implies that $f(x)\le f(y)$ ($f$ is non-decreasing)? – 2012-11-29 -
0The latter case. – 2012-11-29
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0I thought that that was probably the case, since $f[I]$ couldn’t be a closed interval otherwise, but I wanted to make sure, even though it doesn’t actually affect the argument. – 2012-11-29