Interchanging Limits: When Does Equality Hold?

[EV33]

Homework Statement

I was trying to prove something and I ended up in a situation similar to,

(limit t$\rightarrow$0)(limit s$\rightarrow$0) f(x+s,y+t)

=(limit s$\rightarrow$0)(limit t$\rightarrow$0)f(x+s,y+t)

My question is when does this equality hold. I can't find it anywhere?



Thank you.

 

[EV33]

Thank you so much. My function is continuous.

 

[Dick]

Thank you so much. My function is continuous.

Sorry, deleted my answer because I was looking at something else and wanted to concentrate on that for a bit. But sure, if f is continuous at (x,y) you can interchange the limits. I was trying to think of a case where it's not true.

 

[HallsofIvy]

There exist examples, in most Calculus texts, which I don't have available now, of functions f(x, y) in which approaching (0, 0) along any straight line (such as going from (x, y) to (x, 0) then from (x, 0) to (0, 0), which is the same as "lim_(x->0)lim_(y->0) f(x, y)" or going from (x, y) to (0, y) then from (0, y) to (0, 0), which is the same as "lim_(y->0)lim_(x->0) f(x,y)) gives the same answer, the value of the function, so that "situation holds" but taking the limit along a quadratic curve gives a different answer so the function is NOT continuous.