# Set of Points Where f Is Continuous

1. Feb 22, 2009

### e(ho0n3

The problem statement, all variables and given/known data
Let C be the set of points where f: R --> R is continuous. Show that C may be written as the intersection of a countable collection of open sets in R.

The attempt at a solution
If C is empty, the result is true. If C has countably many points, say x_0, x_1, ..., then R - {x_0, x_1, ...} is the union of countable collection of open intervals, (a_i, b_i) and so we may write C as the intersection of {[a_i, b_i]}. Of course, this is not what I want, but it's the best idea I've had so far. I don't know what to do if C is uncountable. Any tips?

2. Feb 22, 2009

### e(ho0n3

I have another idea for the countable case: If C = {x_0, x_1, ...}, for each x_i, we may write

x_i = intersection {(x_i - 1/n, x_i + 1/n) : n = 1, 2, ...}

which is a countable intersection. Thus, C is the intersection of countably many intersections of countably many open intervals.

3. Feb 24, 2009