# Continuously differentiable function that is injective

If $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable function such that $g'(a) \neq 0$ for all $a \in \mathbb{R}$, show that g is injective.

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Suppose there exist $a \neq b$ such that $g(a)=g(b)$. Then use mean value theorem to find $c \in (a,b)$ such that $g'(c)=0$.
@user43418: You can conclude by contraposition, "if $g$ is not injective then $g'$ has a zero" is equivalent to "if $g'$ doesn't vanish then $g$ is injective". –  Seirios Mar 31 '13 at 11:18