Okay, here is one:
Suppose that f:R-->R is continuous for all x and let alpha be in R. Show that if there is a b in R such that f(b)>alpha, then there is a delta>0 so that for all x such that |x-b|<delta, we have f(x)>alpha
So far, I know that you're going to have to make use of the definition of continuity. THat's about as far as I got though.
Another:
Suppose that f:R-->R is continuous for all x and let f(q)=0 for all rationals q. Prove that f(x)=0 for all x