Consider an indexing $q_1,q_2,\dots$ of the rationals in $(0,1)$ and define $f:(0,1)\to (0,1)$ by $f(x)=\sum_{j: q_j < x} 2^{-j}$.
Prove that $f$ is discontinuous at all rationals and continuous at all irrationals.
I tried to prove this by definitions but I failed. For example, what I had for discontinuity:
Let $x_0$ be rational. We need to show that for all $\delta > 0$ there is $x\in (x_0-\delta,x_0+\delta)$ such that $|f(x)-f(x_0)| \ge \epsilon$ for some $\epsilon > 0$. By symmetry we may only look for $x > x_0$, and also $x_0$ must be rational. Let's look at $f(x)-f(x_0)$. We have $$f(x)-f(x_0)=\sum_{j=1}^n 2^{-j}-\sum_{j=1}^m 2^{-j}=\frac{1}{2^{m+1}}+\dots+\frac{1}{2^n};$$ here $q_1,\dots,q_n < x, q_{n+1}\ge x$ and $q_1,\dots, q_m < x_0, x_{m+1}\ge x_0$ with $n \ge m$. The number $x_0$ (and hence $m$) is fixed, and we need to find $x$ such that $$\frac{1}{2^{m+1}}+\dots+\frac{1}{2^n} \ge \epsilon$$ for some $\epsilon > 0$ and $x$ is as close to $x_0 $ as we wish. There are infinitely many rationals in any neighborhood of $x_0$, but how to choose the one we need?
With the continuity part, I also didn't come to anything worthwhile.
Given a rational $r$ there must exist an index $i'$ such that $q_{i'} = r$, take any $x>r$ then,
$$f(x) - f(r ) = \sum_{j:q_j<x} 2^{-j} - \sum_{j:q_j<q_{i'}} 2^{-j} = \sum_{j:q_{i'}\leq q_j<x} 2^{-j} > \sum_{j:q_{i'}= q_j} 2^{-j} = 2^{-i'} $$ hence $|f(x)-f(r)|$ can't be made arbitrarily small, so $f$ isn't right-continuous at any rational $r \in [0,1)$. Note that it is left-continuous at rational $r\in(0,1]$.
On the other hand if $p$ is irrational, given any $0< \epsilon < 1$, there exists some index $N$ so that $\sum_{k=N}^\infty 2^{-k} = 2^{1-N} < \epsilon$, now for consider $x$ such that $$|x-p| < \min\{|q_1-p|, |q_2-p|,\dotsb, |q_N-p|\} \equiv \delta$$ where $\delta>0$ because the minimum is taken over a finite number of terms and $|q_i-p|>0$ because $q_i\neq p$ for any $i$ because $p$ is irrational.
If $x>p$ then $$|f(x)-f(p )| = \sum_{j: q_j < x} 2^{-j} - \sum_{j: q_j < p} 2^{-j} = \sum_{j: p \leq q_j <x} 2^{-j} \overset{(*)}{<} \sum_{N<j}2^{-j} < \epsilon $$ If $x\leq p$ then $$|f(x)-f(p )| = -\left(\sum_{j: q_j < x} 2^{-j} - \sum_{j: q_j < p} 2^{-j}\right) = \sum_{j: x \leq q_j <p} 2^{-j} \overset{(*)}{<} \sum_{N < j}2^{-j} < \epsilon $$
where $(*)$ follows by by the choice of $\delta$, as for any rational $q_j$ satisfying $|q_j - p|< \min\{|q_1-p|,\dotsc,|q_N-p|\}$ it is clear that $j>N$.
Hence $f$ is continuous at irrational $p$.