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## Homework Statement

Show that [itex]f(x) = \sum_{i=1}^{\infty}\frac{2^{i}x - \lfloor 2^{i}x \rfloor}{2^{i}}[/itex] is continuous at all real numbers, excluding integers.

## The Attempt at a Solution

I've tried going about via |f(x) - f(y)| < ε, but am having trouble with this, since first, I don't get anywhere and can't really pin down the floor parts of the function, and second, I'm a bit confused, since I also have to deal with n → ∞, in addition to, say, y → x to show continuity.

What I got this way is [itex]f(y) - f(x) = \sum_{i=1}^{\infty}\frac{2^{i}(y - x) + \lfloor 2^{i}x \rfloor - \lfloor 2^{i}y \rfloor}{2^{i}}[/itex], but after spending hours on this I have zero idea what to do with that.

Any help would be greatly appreciated, as I'm really struggling here.

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