Analysis (of the real variety)?

Question

Anyone able to help me with the following proof?

If f is a continuous function on a compact set, show that either f has a zero or f is bounded away from zero.

thanks

Answer 1

Suppose f is a continuous function on a compact set such that f is not bounded away from zero. Then every open ball surrounding zero contains at least one point of the range of f, so 0 is an accumulation point of the range of f. However, the image of a compact set under a continuous function must be compact, and therefore must be closed (since R is Hausdorff). So it follows that every accumulation point of the range of f must actually be in the rage of f, so 0 is in the range of f, thus f has at least one zero. Q.E.D.