(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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].

2. Relevant 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?

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Uniform convergence and continuity

**Physics Forums | Science Articles, Homework Help, Discussion**