Does integration preserve continuity?

The answer is no.

For instance, note that Fourier analysis tells us that there exists a sum of continuous functions $g_k(x)$ over $[0,1]$ such that the sum $g(x) = \sum_{k=1}^\infty g_k(x)$ is discontinuous (the link is to the Fourier series of a square wave). With that in mind, define $$ f: [0,1]\times[0,1]\to \Bbb R\\ f(x,y) = k(k+1)g_k(y) \quad \text{for all } x \in (1/(k+1),1/k] $$ You will find that $\int_0^1 f(x,y)\,dx = g(y)$

Here is a simple counterexample: Take $f(x,y) = |y|\max(0,1-|xy|)$, then $\int f(x,y) dx = \begin{cases} 1 , & y\neq 0 \\ 0, & y=0 \end{cases}$