I found a certain theorem, that hasn't been proven in the article and as it seems to require just Calculus 2 (multivariable calculus) knowledge.- I would like to prove it, however I am completely stuck how to approach this problem.
Therefore I would be very happy, if someone could provide any sort of constructive hint, comment or answer or recommendation for further reading. As always thanks in advance.
The theorem states as follows:
Let $ f : [a, b]×Y → \mathbb{R} $ be a function, with $ [a, b]$ being a closed interval, and Y being a compact subset of $\mathbb{R}^n$.
Suppose that both $f(x, y)$ and $\frac{∂}{∂x}f(x, y)$ are continuous in the variables $x$ and $y$ jointly.
Then $ \int_Y f(x, y)dy$ exists as a continuously differentiable function of $x$ on $[a, b]$ with:
$\frac{d}{dx} \int_Y f(x, y)dy = \int_Y \frac {∂}{∂x} f(x, y)dy $