# Reversal of the order of limits

1. Jul 12, 2009

### 22/7

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.

2. Jul 13, 2009

### HallsofIvy

Staff Emeritus
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.