I might have made a mistake, but I think f(x)=x^2 works for the second statement but fails for the first. Reasoning below.

Statement (b) means that for all real numbers , the limit exists and equals f(a). Statement (a) on the other hand means that there is a way to choose that relies only on and not on , to satisfy the definition of limit/continuity for all real . (Recall that many epsilon-delta proofs from calculus involve finding an expression for delta in terms of a and epsilon. See here for an example.) Can you see why (a) is stronger and implies (b)?

If I've made a mistake, I'm pretty sure someone will come along and correct me before long.