# Continuous functions proof

1. Dec 9, 2012

### SMA_01

Let f and g be two continuous functions on ℝ with the usual metric and let S$\subset$ℝ be countable. Show that if f(x)=g(x) for all x in Sc (the complement of S), then f(x)=g(x) for all x in ℝ.

I'm having trouble understanding how to approach this problem, can anyone give me a hint leading me in the right direction?

Thank you.

2. Dec 9, 2012

### Dick

How about trying to show that Sc is dense in R? That would do it, yes?

Last edited: Dec 9, 2012
3. Dec 9, 2012

### SMA_01

I was told that Sc is dense because S is countable. I'm not sure if that's a theorem, but should I just prove density using the definition or is there a simpler way?

4. Dec 9, 2012

### Dick

Use the definition. You'll have to add to that what you hopefully know about some subsets of R being uncountable.

Last edited: Dec 9, 2012