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.