Counterexample to switching limit and integral on compact domain If $f:[a,b] \times [c,d] \to \mathbb{R}$ is continuous on the compact rectangle (and, hence, uniformly continuous) then it holds for $y_o \in [c,d]$ that
$$\lim_{y \to y_0}\int_a^b f(x,y) \,dx = \int_a^b \lim_{y \to y_0}f(x,y) \,dx =\int_a^b f(x,y_0) \,dx. $$
I would like to find some counterexamples where the switch is not permitted when, if possible, $f(\cdot,y)$ is integrable over $[a,b]$ and $f(x, \cdot)$ is continuous on $[c,d]$.
I would imagine such a function is not uniformly bounded. Otherwise, I believe the bounded convergence theorem would justify the switch.
Thank you. 

Try $$ f(x,y) = \cases{ \dfrac{x}{y^2} \exp(-x/y) & if $y \ne 0$\cr
                    0 & if $y = 0$ }$$
on $[0,1] \times [0,1]$, with $y_0 = 0$.
