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

  • Thread starter davitykale
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  • #1
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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?
 

Answers and Replies

  • #2
Dick
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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?

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.
 
  • #3
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I think I'm confused...wouldn't that mean that c is continuous when it is an element of the given set?
 
  • #4
Dick
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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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