Density of Countable Sets in ℝ and its Implications for Continuous Functions

[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 S^c (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.

 

[Dick]

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 S^c (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.

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

 

[SMA_01]

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

I was told that S^c 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?

 

[Dick]

I was told that S^c 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?

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