Define a function $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Suppose that for every fixed $x \in \mathbb{R}, f(x,y)$ is continuous. And suppose that for every fixed $y \in \mathbb{R}, f(x,y)$ is integrable. Suppose also that there exists an integrable function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $|f(x,y)| \le g(x)$ for all $x,y$. Show that the function $h : \mathbb{R} \rightarrow \mathbb{R}$ defined by $h(y) = \int_{\mathbb{R}} f(x,y) dx$ is continuous.
So far, all I have really gotten is that, since $f(x,y)$ is continuous for every fixed $x \in \mathbb{R}$, we have that $\forall x \in \mathbb{R}$ and $\forall \epsilon > 0$, $\exists \delta_x > 0 $ such that $| f(x,y) - f(x,z) | < \epsilon$, $\forall y,z \in \mathbb{R}$ with $| y - z | < \delta_x$. So then we have $$ | h(y) - h(z)| = \left| \int_{\mathbb{R}} (f(x,y) - f(x,z)) dx \right| $$ and we can then say something about $f(x,y)$ being continuous for every fixed $x$.
From here I am stuck; the tightest bound I've been getting is an integral over $\mathbb{R}$ of a constant.
Any hints are appreciated!!
Hint -- assuming as stated in the body of the question that $|f(x,y)| \leqslant g(x)$ for all $x$ and $y$ where $g$ is integrable:
Given a fixed $y_0$ and a sequence $y_n \to y_0$, we have by continuity $f(x,y_n) \to f(x,y_0)$. Now apply the dominated convergence theorem to
$$h(y_n) = \int_{\mathbb{R}} f(x,y_n) \, dx$$