Here's the question:
"Let K be a compact subset in and let be a continuous function. Prove that for every there exists such that
for every ."
I don't really see the "point" of the question - can't we just take ? What are we "supposed" to do?
Here's the question:
"Let K be a compact subset in and let be a continuous function. Prove that for every there exists such that
for every ."
I don't really see the "point" of the question - can't we just take ? What are we "supposed" to do?
You can't just take , because has to be a constant. Suppose for example that f is the function defined on the interval K=[0,1]. Then becomes unbounded when x=0 and y→0.
I think the trick is to consider two separate cases: when x and y are close together, and when they are not.
Given , the uniform continuity of f tells you that there exists a such that whenever . That deals with the case when x and y are close together (in other words, within δ of each other). Now you just have to show that is bounded when . (I'll leave that part to you.)