- #1
Poopsilon
- 294
- 1
So my text says that the piece-wise function f(x,y)=(x^3)/(x^2 + y^2) for (x,y)≠(0,0) and f(0,0)=0 is continuous due to the fact both its partial derivatives are bounded. I can't see how this follows, I mean the only point at which the reader would be worried about continuity is at the point (0,0) and to show that it is continuous at that point I would simply bound the function by g(x,y)=x and then show that this function goes to 0 as x -> 0. But why would the partial derivatives being bounded allow us to conclude that f(x,y) is continuous at (0,0)?