# Proving discontinuities

1. Jul 5, 2009

### foxjwill

1. The problem statement, all variables and given/known data
Find (and prove as such) all discontinuities of the function $$g:[0,1]\to\mathbb{R}$$ given by
$$g(x)=\sum_{n=1}^\infty \frac{1}{2^{2n-1}}\left\lfloor \frac{2^nx+1}{2} \right\rfloor$$​
where $$\lfloor\cdot\rfloor$$ is the greatest integer function.

2. Relevant equations

3. The attempt at a solution
I'm pretty sure that the discontinuities all occur at $$x=(2k+1)2^{-m}$$ for positive integer $$k,m$$ since this is where the expression inside the greatest integer function is an integer. The thing is, I have no how to go about proving that these points are discontinuous. Can anyone steer me in the right direction?

2. Jul 5, 2009

### Office_Shredder

Staff Emeritus
When x hits that point, note that every term in the sum is
1) positive
2) larger when you cross over the point

And you get that as you approach (2k+1)2-m from the right, each term is strictly larger than when you approach from the left