Doubt: Cauchy Equation over real

Question

I was going through Evan Chen's Functional Equation handout handout

I couldn't under stant the second and the third bulleted points , can someone explain me ?

Answer 1

A function is bounded above if there exist $M$ such that $\forall x \in I ,(I \subseteq R)$ then, $$f(x) \le M $$ on the other hand if we have $ \ge $ in the equation then it's called bounded below

Answer 2

A real valued function $f$ is bounded above on a set $S$ is there exist $M$ such that $f(x) \leq M$ for all $x \in S$. It is bounded below on $S$ is there exist $m$ such that $f(x) \geq m$ for all $x \in S$. It is bounded if it is bounded above and below.

A non trivial interval is an interval of one of the following types where $a<b$ and $c \in \mathbb R$:

$(a,b),[a,b),(a,b], [a,b], (-\infty, \infty), (c,\infty), [c,\infty)$, $(-\infty, c), (-\infty, c]$.