Composition of continuous and Lp function is essentially bounded? Let $f:\Omega \rightarrow \mathbb{R} $ be a function in $L^p(\Omega)$ for some $1\leq p < \infty$ and $g: \mathbb{R} \rightarrow[a,b]$ continuous, with $a,b\in \mathbb{R}$. 
Do we have $g \circ f \in L^\infty(\Omega)$? Since $\sup\limits_{\mathbb{R}}g <\infty $ we should have $\| g \circ f \|_{L^\infty(\Omega)} < \infty$ so the answer should be yes, but I have the feeling I'm missing something here.
Thank you in advance for any hint.

Of course.  For all $t\in\mathbb{R}$, $g(t)\in [a,b]$, so for all $x\in\Omega$, $g(f(x))\in[a,b]$.  So $g\circ f$ is indeed a bounded function.
