f: [-1, 1]---R and the graph of f is compact, prove that f is continuous?
Any hint will be appreciated!!
Assume that the graph is compact. Assume for contradiction that is discontinuous at . Then there exists an and a sequence such that converges to . Look at the sequence . Since the graph is compact it has a subsequence which converges to a point . So converges to . But converges to which is a contradiction. This implies that is continuous and onto.
This is my understanding. By "graph" the poster means "image".
So the poster is asking: given a function mapping a compact set ([-1,1] is compact) into a compact set does it mean that the function is continous on the set?
That is what I did.
Now, I gave an example involving the Dirichelt (discontinous) function which shows it is false. You are saying you proved it, if so, then why do I have a conter-example?