proving a differentiable function $f: \Bbb R \to \Bbb R$ and a constant $c>0$ with $f'(x) \geq c$ for all $x \in \Bbb R$. is a bijection We have a differentiable function $f: \Bbb R \to \Bbb R$ and a constant $c>0$ with $f'(x) \geq c$ for all $x \in \Bbb R$.
Show that $f$ is a bijection from $\Bbb R$ to $\Bbb R$.
From Rolle's theorem follows that if $f'(x) \neq 0$  $f$ is injective. Which is the case here so we know that $f$ is injective.
I'm stuck on proving it's surjective.
In the questions before this one I already proved with the mean value theorem that $f(x) \geq f(0)+ cx$ if $x \geq 0$, and that  $f(x) \leq f(0)+ cx$ for all $x \leq 0$

From $c>0$ and $f(x) \geq f'(0)+ cx$ for $x \ge 0$ we get
$ \lim_{x \to \infty}f(x)= \infty$.
From $c>0$ and $f(x) \le  f'(0)+ cx$ for $x \le 0$ we get
$ \lim_{x \to -\infty}f(x)= -\infty$.
Now we derive , by the intermediate value theorem: $f( \mathbb R)= \mathbb R$
