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.


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 $g'(c)=0$.

  • $\begingroup$ I think we don't need to use the mean value theorem. Since g(a)=g(b), we can use Rolle's theorem $\endgroup$ – user43418 Mar 31 '13 at 11:12
  • $\begingroup$ @user43418: You are right; in fact, the two theorems are equivalent. $\endgroup$ – Seirios Mar 31 '13 at 11:13
  • $\begingroup$ But I still don't know how to conclude $\endgroup$ – user43418 Mar 31 '13 at 11:15
  • $\begingroup$ @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". $\endgroup$ – Seirios Mar 31 '13 at 11:18

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