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

[davitykale]

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?

 

[Dick]

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.

 

[davitykale]

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

 

[Dick]

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.