# Math Help - uniform convergence

1. ## uniform convergence

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...

2. Originally Posted by jimmybuffet
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.
If $f_n: X\mapsto Y$ are all continous.
Consider,
$|f(x) - f(x_0)| \leq |f(x) - f_n(x)|+|f_n(x)-f_n(x_0)|+|f_n(x_0)-f(x_0)|$
All three can be made small.

3. thank you, that is helpful. btw either you're really good at pacman, or you're the only one who plays it. jk nice awards there.

4. ## yeah...

well i still haven't figured out how to use the idea of compactness here, so maybe i'll just have to face up to all the evidence and admit that i am an idiot--anyway, can anyone help?

5. Originally Posted by jimmybuffet
well i still haven't figured out how to use the idea of compactness here, so maybe i'll just have to face up to all the evidence and admit that i am an idiot--anyway, can anyone help?
Just note that $\left[ {x_0 - 1,x_0 + 1} \right]$ is a compact subset of $\Re$.

6. ## okay

it is actually supposed to be f: X to Y where X and Y are arbitrary metric spaces, but since R isn't compact, i'll try to generalize from what you say (closed neighborhood or something? but not every closed and bounded set is compact i am told)