TITLE: Change of order of double limit of function sequence
QUESTION [10 upvotes]: The more general quesion is under what conditions the folloing equality will hold (all functions are real valued):
$$\lim_{x \rightarrow a} \ \lim_{j \rightarrow \infty} f_j(x) = \lim_{j \rightarrow \infty} \ \lim_{x \rightarrow a} f_j(x)$$
A more specific question is if it will hold for non-continuous functions $f_j$ that are uniformly convergent to a non-continuous limit function $f$ and all the single and iterated double limits exist.
Also some references would be useful.

REPLY [2 votes]: Note that 
$$\left |\lim_{x\to a}f_j(x) - \lim_{x\to a} f(x)\right|\le \|f_j-f\|_\infty\to 0,\; j\to \infty$$
Thus $$\lim_{j\to\infty}\lim_{x\to a}f_j(x) = \lim_{x\to a} f(x) =  \lim_{x\to a} \lim_{j\to\infty}f(x)$$