Can you give me an example of a function that is either upper OR lower quasi-continuous but not both?

Question

A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty open set $G \subset U$ such that $f(y)< f(x) + \epsilon$ ($f(y)> f(x) - \epsilon$), for each $y \in G$.

Answer 1

Let

$$ f(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases} $$

Then, at $x_0 = 0$, for any $\varepsilon > 0$, take any neighborhoods/open sets $G,U$, and you'll have

$$ f(y) < f(x_0) + \epsilon = 1 + \epsilon $$

for all $y \in G$.

On the other hand, if you changed the definition of the function to $$ f(x) = \begin{cases} 0 & x \leq 0 \\ 1 & x > 0 \end{cases} $$ then for $\epsilon < 1$, any $y > 0$ will fail $$ f(y) < f(x) + \epsilon = 0 + \epsilon < 1 $$

(you can check that this function is now lower-semicontinuous though)