True or false: If it's true, give an example. If it's false, prove it.

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Homework Help Overview

The discussion revolves around the continuity of a function defined from the real numbers to the real numbers, specifically examining the conditions under which a function is continuous at a point c in relation to a particular set of numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the idea of defining a piecewise function to demonstrate continuity and question whether continuity can be established at irrational numbers. There is also a challenge regarding the density of the defined set and its implications for continuity.

Discussion Status

Participants are actively questioning the assumptions surrounding the continuity of the function at points in the specified set. Some have offered examples and counterexamples to illustrate their points, while confusion remains about the implications of the set on continuity.

Contextual Notes

There is a focus on the definition of continuity and the specific characteristics of the set { m/2^n: m,n in Z, n>=0}, with participants noting its density and its effect on nearby points.

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Homework Statement


A function f: R -> R such that f is continuous at a point c if and only if c is not an element of the set: { m\2^n: m,n in Z, n>=0)


Homework Equations


Definition of continuity/discontinuity?


The Attempt at a Solution


Is it enough to say that we can define a piecewise function where f(x) = 0 if x is an element of the described set, and f(x) = x otherwise...then we maybe have continuity at points such as irrational numbers?
 
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davitykale said:

Homework Statement


A function f: R -> R such that f is continuous at a point c if and only if c is not an element of the set: { m\2^n: m,n in Z, n>=0)


Homework Equations


Definition of continuity/discontinuity?


The Attempt at a Solution


Is it enough to say that we can define a piecewise function where f(x) = 0 if x is an element of the described set, and f(x) = x otherwise...then we maybe have continuity at points such as irrational numbers?

I would be if you could prove it. But you can't. The set you've defined is dense. It doesn't work. Try f(m/2^n)=1/2^n for points in the set and zero otherwise.
 
I think I'm confused...wouldn't that mean that c is continuous when it is an element of the given set?
 
davitykale said:
I think I'm confused...wouldn't that mean that c is continuous when it is an element of the given set?

No, it's not continuous at points of the form m/2^n. There's an irrational in every neighborhood of one of those points.
 

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