Prove or disprove the following statement:
There exist functions $f,g:\mathbb{R}→\mathbb{R}$ such that:
(1) $f,g$ are semi-continuous, i.e. for each $x\in\mathbb{R}$, $f$ is either upper semi-continuous at $x$ or lower semi-continuous at $x$.
(2) $f+g$ is nowhere semi-continuous, i.e. for each $x\in\mathbb{R}$, $f$ is neither upper semi-continuous at $x$ nor lower semi-continuous at $x$.
I read a result on a book constructing three semi-continuous functions whose sum is nowhere semi-continuous. This lead me to start thinking whether two functions would be enough, but I'm getting nowhere with it. Any thought would be helpful.
Here is the three-function construction from the book I was reading: $$f_{1}(x) = \begin{cases} \frac{4}{2q+1} & \text{ if } x=\frac{p}{2q+1}; \\ -2-\frac{4}{2q}& \text{ if } x=\frac{p}{2q}; \\ -2 & \text{ if } x \text{ is irrational}. \end{cases}$$ $$f_{2}(x) = \begin{cases} -1-\frac{1}{2q+1} & \text{ if } x=\frac{p}{2q+1}; \\ 1+\frac{1}{2q}& \text{ if } x=\frac{p}{2q}; \\ -1 & \text{ if } x \text{ is irrational}. \end{cases}$$ $$f_{3}(x) = \begin{cases} -1-\frac{1}{2q+1} & \text{ if } x=\frac{p}{2q+1}; \\ 3+\frac{1}{2q}& \text{ if } x=\frac{p}{2q}; \\ 3 & \text{ if } x \text{ is irrational}. \end{cases}$$