- #1

complexnumber

- 62

- 0

## Homework Statement

Theorem:

Let [tex](X,d_X),(Y,d_Y)[/tex] be metric spaces and let [tex]f_k : X \to Y[/tex], [tex]f :

X \to Y[/tex] be functions such that

1. [tex]f_k[/tex] is continuous at fixed [tex]x_0 \in X[/tex] for all [tex]k \in \mathbb{N}[/tex]

2. [tex]f_k \to f[/tex] uniformly

then [tex]f[/tex] is continuous at [tex]x_0[/tex].

## Homework Equations

If all [tex]f_k[/tex] are continuous on [tex]X[/tex] and [tex]f_k \to f[/tex] pointwise, then [tex]f[/tex] 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?