Say a differentiable function $f: I \rightarrow \mathbb{R}$ is defined on an open interval $I$, and there is a closed bounded interval $[a,b] \subseteq I$. If $f'$ is injective on $[a,b]$, is it continuous on $(a,b)$?
2026-05-14 07:37:26.1778744246
Is a function's derivative continuous on $(a,b)$ if it is injective on $[a,b]$?
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Yes. It's a consequence of Darboux theorem, according to which the derivative $f'$ of a differentiable function $f:I\to \Bbb R$ has the intermediate value property (IVP). Since a real-valued function $g$ defined on some interval $J$ is a homeomorphism onto its image if and only if it is injective and it has the IVP, your $f'$ will be continuous.