If somebody comes up with a better title, then please feel free to tell me.
Let $(\Omega,\mathcal F,\mu)$ be a measure space. Let $(a\,..b)\subseteq \mathbb R$ and $f : (a\,..b) \times \Omega \to \mathbb R$ be such that, $f(x,\_)$ (*) is $\mu$-integrable for all $x$ and $f(\_,\omega)$ is continuous for all $\omega$. Let $g : \Omega \to \mathbb R$ be $\mu$-integrable such that $|f(x,\_)| \leq g$ for all $x\in \Omega$.
Then one can show that $x\mapsto F(x):=\int f(x,\_) d\mu$ is continuous.
This was an exercise with a hint: "Apply dominated convergence theorem for a suitable sequence of functions."
My question is: How do I prove this? I'm completely stuck since I have no idea what kind of sequence I'm supposed to look at.
(*) The symbol $\_$ is a placeholder for an argument, e.g. $f(x,\_)$ is a function $\Omega \to \mathbb R$.