I essentially need to prove Leibniz's Rule for differentiation under an integral: given a continuous function $f(x,y)$ and a continuous partial $\frac{\partial f}{\partial y}$ prove that if $G(y) = \int_a^b f(t,y) \, dt$ then $G'(y) = \int_a^b \frac{\partial f}{\partial y}(t,y) dt$.
There are lots of proofs for this online, but I can't quite understand why they require the partial derivative to be continuous. As far as I can see, I need to prove that
$\lim_{h \to 0} \frac{G(y+h)-G(y)}{h} = \lim_{h \to 0} \int_a^b \frac{f(t,y+h)-f(t,y)}{h} dt = \int_a^b \frac{\partial f}{\partial y}(t,y) dt$ which, using the epsilon-delta definition of the limit means that I want to show that for any fixed $\epsilon$ I can find a $\delta > 0$ such that as long as h < \delta \left| \int_a^b \frac{f(t,y+h)-f(t,y)}{h} - \frac{\partial f}{\partial y}(t,y) dt \right| < \epsilon At this point most of the proofs I've seen turn to the Mean Value Theorem, and then make use of the continuity assumption, but doesn't the existence of $\frac{\partial f}{\partial y}$ mean that $\lim_{h \to 0} \frac{f(t,y+h)-f(t,y)}{h} = \frac{\partial f}{\partial y}$ which means it's relatively easy to select a relevant $\delta$ without having to turn to continuity? Sorry if that was overly verbose, and thanks for any help!!