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If $f$ is continuous on $[a,b]$ and $f\;'> 0$ on $(a,b)$, then $f$ is monotonically increasing on $[a,b]$ and proof follows from mean value theorem. Now suppose we have $f \in C[a,b]$ and $f\;'(x+)> 0$ for all $x \in (a,b)$. Does it follow that $f$ is monotonically increasing on $[a,b]$?

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