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!

Here is how you advance. We need to prove that

$\forall \epsilon >0 $ thers exists $\delta$ such that
$$ |h|<\delta \implies |g(x+h) - g(x)| < \epsilon $$

We advance as
$$ g(x+h) - g(x)= \int\limits_{c}^{d} f(x+h,y) dy - \int\limits_{c}^{d} f(x,y) dy $$
$$  = \int\limits_{c}^{d} (f(x+h,y)  - f(x,y)) dy $$
$$ \implies |g(x+h) - g(x)| =\Bigg| \int\limits_{c}^{d} (f(x+h,y)  - f(x,y)) dy \Bigg| $$
$$\implies |g(x+h) - g(x)| \leq  \int\limits_{c}^{d} |f(x+h,y)  - f(x,y)| dy. $$
You need to use the fact that the function $f(x,y)$ is continuous on $[a,b]\times [c,d]$ to finish the problem.
