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.)