Composing a continuous function with an $L^p$ function.

Question

Let $U \subset \mathbb{R}^n$ be bounded and $1 \leq p < \infty$. Assume $u \in L^p(U)$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Is it true that $f(u) \in L^p(U)$?

I think it's true but I can't seem to find a theorem that talks about it. Is there a name for this result if it is true?

Answer 1

No, this is not true. Pick any $u \in L^1(U) \setminus L^2(U)$, such as $u(x)=|x-x_0|^{-n/2}$ (with $x_0\in U$) and take $f(x)=x^2$.

It is true, however, if $f$ is a bounded function of more generally if $|f(x)|\leq C|x|+b$ for some constants $C,b$. -- PhoemueX