Functions whose derivatives grow fast

If $c<0$, the exponential function is a solution.
If $c=0$ (also true if $c<0$), $x\longmapsto exp(2x)$ works.
If $c>0$, it is impossivle : let $g$ be $f'-f$.

We have for every real $x$ : $g(x)>c$ and $g'(x)>c$.

But by integrating the second inequality for $x<0$ we have : $g(x)<cx+g(0)$ so that $g(x)<0$ when $x$ approaches negative infinity.

That is contradictory with the first condition.

Note that if $f''$ is not regular enough, we cannot integrate $g'$ but we can bypass this issue by using the Mean Value Theorem.