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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.
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.