# Pointwise limit.

1. Apr 14, 2010

### forty

The set of rational numbers Q is countable, and be therefore be expressed as a countable union Q = Un>=1{rn}. Let R be a metric space with the usual distance. Now define a function f:R->R by setting

f(x) = 1/n if x = rn and 0 if x is irrational

(a) Show that f is continuous at each irrational point and discontinuous at each rational point.

I did this using epsilon-delta but I thought this was pretty easy to see anyway, unless I'm completely wrong.

(b) Show that f is the pointwise limit of a sequence of continuous functions. I.e. find a sequence of continuous functions fn such that for ever x in R then

f(x) = lim(n->inf) fn(x)

This is where I'm stuck and have no idea. Any pointers with this would be very helpful.