# Uniform convergence and continuity

## Homework Statement

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 $$x_0$$.

## Homework Equations

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

## 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?

Related Calculus and Beyond Homework Help News on Phys.org
Dick
Homework Helper
f_k(x)=x^k on [0,1]. What's the limit f(x)?

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