Definition of monotone increasing:
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is monotone increasing (respectively, monotone decreasing) if for any $x_{1}, x_{2} \in \mathbb{R}$ with $x_{1} \leq x_{2},$ it is true that $f\left(x_{1}\right) \leq f\left(x_{2}\right)$ (respectively, $\left.f\left(x_{1}\right) \geq f\left(x_{2}\right)\right)$.
Definition of Right-Continuous:
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is right-continuous at a point $x \in \mathbb{R}$ if for every sequence $\left(x_{n}\right)_{n \in \mathbb{N}} \subset(x, \infty)$ such that $x_{n} \geq x_{n+1}$ for all $n \in \mathbb{N}$ and $\lim _{n \rightarrow \infty} x_{n}=x,$ it is true that $\lim _{n \rightarrow \infty} f\left(x_{n}\right)=x .$
Let, the c.d.f. of a $\operatorname{Unif}((a, b))$ -distributed random variable is $$ F_{X}(x)=\left\{\begin{array}{ll} 0, & x \in(-\infty, a] \\ \frac{x-a}{b-a}, & x \in(a, b) \\ 1, & x \in[b, \infty) \end{array}\right. $$
How to prove that the above CDF (Cumulative distribution functions) is right continuous and monotone increasing? Is the c.d.f. FX also left-continuous?