Let $a$ and $b$ be real numbers with $a<b$. Use the floor and / or ceiling functions to express the number of integers $n$ that satisfy the inequality $a<n<b.$

Let $a$ and $b$ be real numbers with $a<b$. Use the floor and / or ceiling functions to express the number of integers $n$ that satisfy the inequality $a≤n≤b$.

The function INT is found on some calculators, where INT$(x)$ = $\left \lfloor x \right \rfloor$ when $x$ nonnegative real number and INT$(x)$ = $\left \lceil x \right \rceil$ when x is a negative real number. Show that this INT function satisfies the identity INT$(-x)$=$-$ INT$(x)$

Prove that if $x$ is a reall number , then $\left \lfloor -x \right \rfloor = - \left \lceil x \right \rceil$ and$\left \lceil -x \right \rceil = -\left \lfloor x \right \rfloor$