Let $f$ be a real-valued, differentiable and monotonic function on $ [a,b] $. How can i prove (or disprove) continuity of $ f' $ on $ [a,b] $?
2026-03-29 10:15:29.1774779329
If $f$ is differentiable and monotonic on $ [a,b] $, is $f'$ continuous on $ [a,b] $?
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Let's look at a classic example of a function with non-continuous derivative: $g(x)=x^2\sin(1/x)$ (with $g(0)=0$). Then $g'(x)=-\cos(1/x)+2x\sin(1/x)$ which does not tend to $g'(0)=0$. Of course $g$ is not monotone. But $g'$ is bounded as $x\to0$. Define $f(x)=g(x)+Cx$ for some $C$ large enough ($C=2$ will do). In some interval containing zero $f'>0$ but $f'$ is not continuous.