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

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Homework Help Overview

The problem involves two continuous functions defined on the real numbers and a countable subset of the reals. The task is to show that if the functions are equal on the complement of this countable set, they must be equal everywhere on the real line.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the concept of density in relation to the complement of the countable set and question whether this is a theorem or needs to be proven. There is also consideration of using definitions to establish the density of the complement.

Discussion Status

The discussion is ongoing, with participants exploring the implications of the density of the complement of the countable set. Some guidance has been offered regarding the use of definitions and the nature of subsets of the reals.

Contextual Notes

There is uncertainty regarding the classification of the complement of a countable set and its properties, as well as the need to clarify whether certain statements are theorems or require proof.

SMA_01
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Let f and g be two continuous functions on ℝ with the usual metric and let S\subsetℝ 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.
 
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SMA_01 said:
Let f and g be two continuous functions on ℝ with the usual metric and let S\subsetℝ 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.

How about trying to show that Sc is dense in R? That would do it, yes?
 
Last edited:
Dick said:
How about trying to show that Sc is dense in R? That would do it, yes?

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?
 
SMA_01 said:
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?

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

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