calculus - prove that there exists a bijective function

Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with bounded derivative. Prove that there exists a real non-zero constant $c$ such that $g(x)=x+cf(x)$ is a bijective function with differentiable inverse.

Answer

If $|f'|\leq L$, any $c\in\left(0,\frac{1}{L}\right)$ work, since in such a case $g(x)$ is an increasing differentiable function having a derivative bounded away from zero, and such that $\lim_{x\to \pm\infty} g(x)=\pm\infty.$