An increasing function continuous at irrational points, but discontinuous at rational points.

Question

Denote all rational numbers in $(a,b)$ by ${r_n}$. Let $$\sum^\infty_{n=1}C_n< + \infty, \quad C_n>0 \quad(n=1,2,\cdots).$$ Define a function on $(a,b)$ $$f(x)=\sum_{r_n<x}C_n \quad \text{(summation over index $n$ for which $r_n<x$)}. $$ Then $f(x)$ is an increasing function continuous at irrational points, but discontinuous at rational points.

Sorry for my ignorance, I don't understand why "$f(x)$ is continuous at irrational points, but discontinuous at rational points", since between every two points there is a rational point. From where I can see, the function is discontinuous at every point.

Answer 1

It is the other way around: continuous at irrational points.
For $x_0\in\Bbb Q$, $$\lim_{x\to x_0^-} f(x)=\sum_{r_n<x_0}C_n<\sum_{r_n\le x_0}C_n=\lim_{x\to x_0^+}f(x)$$ but for $x_0\notin\Bbb Q$, the two limits are the same.