How to prove that this function is continuous? If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on the rectangle $R=[a,b] \times [c,d]$, prove that the function
$g(x) := \int\limits_{c}^{d} f(x,y) dy$
is continuous on $[a,b]$.
Thanks in advance!

Let $x \in [a,b]$, then $|g(x) - g(t)| \leq \int_c^d |f(x,y) - f(t,y)| dy$.  Now given $\epsilon > 0$ there exists a $\delta > 0$ such that $|(x,y) - (t,x)| < \delta$ then $|f(x,y) - f(t,y)| < \frac{\epsilon}{d-c}$.
