problem: if a sequence of continuous functions fn from X to Y converges uniformly to f on every compact subset of X, i need to show that f is continuous.

it might be useful to use the fact that given a point x0, if for every sequence xk converging to x0 we have f(xk) converges to x0, then f is continuous.

also, if a metric space is X is compact itself, then f is continuous under the conditions in the problem (but of course X can't be assumed to be compact in the problem)

anyway, i just need a place to start because i really don't know what to make of the whole "every compact subset" thing--does it mean that there is potentially a separate delta (or N) for each compact subset or that the same delta may be used for any compact subset of X? i am confused...