Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$. If $f:\Bbb{R}\to\Bbb{R}$ is continuous and $|f(x)|\leq a+b|x|^\alpha$, is the map $F:L^p(\Omega)\to L^{p/\alpha}(\Omega)$ given by $F(u) =f\circ u$ continuous when $p>\alpha$?
I know that that $F$ is well defined since $$\|F(u)\|_{p/\alpha}\leq \|a+b|u|^\alpha\|_{p/\alpha}\leq a|\Omega|^{\alpha/p}+b\|u^\alpha\|_{p/\alpha}= a|\Omega|^{\alpha/p}+b\|u\|_{p}^{\alpha}<\infty.$$
However, I don't know how to invoke the continuity of $f$ in the proof of continuity of $F$, if this is even possible to begin with.
EDIT: fixed a few typos with the exponents.