real analysis - Cumulative Distribution Function that is Discontinuous at All Rationals

I'm completely lost about how to construct this function. I write out the definition to be that it must be some function $F: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies:

(1) $F$ is non decreasing

(2) lim$_{x \rightarrow \infty} F(x) = 1$, lim$_{x \rightarrow -\infty} F(x) = 0$

(3) $F$ is right continuous

(4) $F$ is continuous at all irrationals and discontinuous at all rationals

I need something that has "large jumps" to the left and "small jumps" to the right of every rational to make it right continuous but not left continuous at those points...but then how can it be both left and right continuous at every irrational? I'm horribly confused. What I thought about so far (with the help of other stackexchange questions) is a function like this:

$\sum_{i=1}^{\infty} 2^{-i} \mathbb{I}(x-q_i > 0)$, where $\{ q_i \}_{i \in \mathbb{N}}$ are the set of rational numbers and $\mathbb{I}$ is the indicator function.

Can someone help me prove the properties, if this function satisfies them?