Leibniz Rule (derivative of an integral)

[Castilla]

The Leibniz rule:

1. Let f(x,y) be a continuous two variable real function defined on (closed intervals) {x0, x1} x {y0, y1}.
2. Let f_1 (partial derivative of f with respect to the first variable) exists and be continuous on the same subset of RxR.
3. Let F be defined as F(x) = (int) (lim x = y0, x = y1) f(x,y)dy

Then F' exists and F'(x) = (int) (lim x = y0, x = y1) f_1(x,y)dy.

I know and understand the proof that uses the Mean Value Theorem of elementary calculus. Yet in Wikipedia (Leibniz's rule: derivatives and integrals) they use a simpler proof. They arrived to this point, which I understood:

F'(y) = lim (h->0) (int)(lim x=c, x=d) ( f(x+h, y) - f(x,y)) dy
h

BUt then they say: "... and using uniform continuity the right hand side equals to

(int) (lim x=c, x=d) f_1(x,y)dy."

How do they manage to use uniform continuity to introduce the limit inside the integral ?

Thanks.

 

[fresh_42]

The question is: When are Riemann integrability and limits allowed to be exchanged?
The answer is: If everything exists (integral, limits) and the limit is uniformly, here in the second variable.

https://pdfs.semanticscholar.org/d730/9dc4faee1d6227a896a94c82913c0d2a0a71.pdfhttp://cecas.clemson.edu/~petersj/Courses/M454/Lectures/L23-DiffIntInterchange.pdfhttps://pnp.mathematik.uni-stuttgart.de/iadm/Weidl/analysis2/vorlesung-ana2/node73.html