Homework Help: Uniform convergence and continuity

1. Apr 2, 2010

complexnumber

1. The problem statement, all variables and given/known data

Theorem:

Let $$(X,d_X),(Y,d_Y)$$ be metric spaces and let $$f_k : X \to Y$$, $$f : X \to Y$$ be functions such that
1. $$f_k$$ is continuous at fixed $$x_0 \in X$$ for all $$k \in \mathbb{N}$$
2. $$f_k \to f$$ uniformly
then $$f$$ is continuous at $$x_0$$.

2. Relevant equations

If all $$f_k$$ are continuous on $$X$$ and $$f_k \to f$$ pointwise, then $$f$$ need not be continuous. Why?

3. The attempt at a solution

I really can't think of an example. Can someone please explain to me why this is so or give me an example?

2. Apr 2, 2010

Dick

f_k(x)=x^k on [0,1]. What's the limit f(x)?

3. Apr 2, 2010

complexnumber

The limit $$f(x)$$ is
\begin{align*} f(x) =& 0 \text{ if } 0 \leq x < 1 \\ f(x) =& 1 \text{ if } x = 1 \end{align*}
which is not continuous.

But every $$f_k(x) = x^k$$ is continuous and converges pointwise to $$f$$.

Thank you so much. I never thought about functions like this.