Challenging Compactness/Continuity Problem

[SpaceTag]

Can anyone provide any ideas or hints for this problem?

Let f:R^2 -> R satisfy the following properties:

- For each fixed x, the function y -> f(x,y) is continuous.

- For each fixed y, the function x -> f(x,y) is continuous.

- If K is a compact subset of R^2, then f(K) is compact.

Prove that f is continuous.

 

[Dick]

That is challenging. I've given it some thought and I don't have the answer. But I'll give you a way to start thinking about it. Start trying to find examples of functions that satisfy the first two criteria but which are not continuous and figure out how they violate the third criterion. I'll get you started. f(x,y)=(xy)/(x^4+y^4) and define f(0,0)=0. That's continuous along lines of constant x and y, but f is discontinuous at (0,0) and f(K) for K a compact set containing an open neighborhood of the origin is not compact. Because f(x,y) is unbounded near (0,0).