The inverse function theorem or implict function

Question

Let $U \subset \mathbb{R}^n$ be an open set and $f:U \to \mathbb{R}$ a continuous function.

The function $g:U \to \mathbb{R}$ is given by $$g(x)=\int_{0}^{f(x)}\!\left(t^2+1\right)\mathrm dt$$

If $g$ is in $C^1$ then is $f$ in $C^1$ too?