Totally Bounded in a Function Space

[jdcasey9]

Homework Statement

Let (X,d), (Y,p) be metric spaces, and let f,fn: X -> Y with fn->f uniformly on X. Show that D(f)c the union of D(fn) from n=1 to n=infinity, where D(f) is the set of discontinuities of f.

Homework Equations

The Attempt at a Solution

Ok, so this looks pretty close to just a straight forward problem of total boundedness, but I'm assuming there is some difference that I'm overlooking. If we just try to show that D(f) is totally bounded we get:

Let E>0. Take B(E, f1), ... , B(E, fn) (where E is the radius and f is the center) over the interval f = (f1,fn) where fi+1 - fi = E/2. Therefore, we have an E-net covering D(f) and D(f) is totally bounded.

Is there something more to show/do or am I way off?

 

[micromass]

Ummm, what does total boundedness has to do with this?

Cant you just show:
If x is a point such that all f_n is continuous in x, then f is continuous in x.

That would prove that

$$\bigcap_n{X\setminus D_{f_n}}\subseteq X\setminus D_f$$

 

[jdcasey9]

Really? Are you sure? What we are showing matches my definition of totally bounded nearly verbatim.