When Is the Reversal of Limits Valid?

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When is the aforementioned reversal valid? (For example, is continuity in the neighborhood of the limit sufficient?)

Also, how would one prove this? In the example that I am analysing (that of the calculation of a second partial derivative), the function has a removable discontinuity at the point of the limit and also along both lines leading to it if the limits are taken indepedantly (one then the other).

As I write thoughts of a possible proof form in my mind but any input would be much appreciated.

 

[HallsofIvy]

Generally speaking, if the convergence is uniform, you can swap limits. Actually, when it is possible to interchange limits is one of the major questions of Mathematical Analysis.