Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function and define the functional $F:\mathbb{R}\to\mathbb{R}$ by $F(t)=\int_{0}^{t}f(s)\,ds$.
Now define the functional $G:W_{0}^{1,p}(\Omega)\to\mathbb{R}$ by $G(u)=\int_{\Omega}F(u)\,dx.$
Then $G$ is Frechet differentiable and $<G'(u),\phi>=\int_{\Omega}f(u)\phi\,dx$ for all $\phi\in W_{0}^{1,p}(\Omega)$.
I have checked that $G$ is Gateaux differentiable in the sense that $$\lim_{t\to 0}\frac{G(u+t\phi)-G(u)}{t}=\int_{\Omega}f(u)\phi\,dx=<G'(u),\phi>.$$
Now to prove $G$ is Frechet differentiable which is generally referred as $C^1$, we need to prove that $G'$ is continuous, that is for every $u_n\to u$, one has $G'(u_n)\to G'(u)$.
First of all it is not clear to me, whether for such continuous $f$, $G'$ is always continuous.
If somebody kindly clarify the same, it would help me a lot.