f: [-1, 1]---R and the graph of f is compact, prove that f is continuous?

Any hint will be appreciated!!

Printable View

- Oct 11th 2007, 06:03 PMvioletsfadvanced calculus, help!
f: [-1, 1]---R and the graph of f is compact, prove that f is continuous?

Any hint will be appreciated!! - Oct 11th 2007, 06:43 PMtukeywilliams
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.

- Oct 11th 2007, 06:53 PMThePerfectHacker
- Oct 11th 2007, 07:02 PMtukeywilliams
I think I proved the right statement?

- Oct 11th 2007, 07:08 PMvioletsf
- Oct 11th 2007, 07:53 PMThePerfectHacker
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? - Oct 11th 2007, 07:59 PMtukeywilliams
is continuous on if and only if is compact. I proved the direction.

I defined - Oct 11th 2007, 08:05 PMThePerfectHacker