I proved that if $\mathit\\f$ is an increasing function in ($\mathit\\a,b$) then $\lim \limits_{x \to \mathit\\a+}f(x)$ exist and equals to $\inf\{f(x) | a \lt x \lt b\}$.
But is it also true that if $\mathit\\f$ is increasing in [$\mathit\\a,b$) then $\lim \limits_{x \to \mathit\\a+}f(x)$ = $\inf\{f(x) | a \le x \lt b\}$?
Consider $$f(x) = \begin{cases} x & x\in(0,1) \\ -10 & x=0 \end{cases} $$ Then, $\lim_{x\to 0^+}f(x)=0$.
Furthermore, $\lim_{x\to a^+}f(x)=\inf\{f(x)|a<x<b\}$ is true if we allow $\inf\{f(x)|a<x<b\}$ to be $-\infty$. If we don't allow this, then the assertion is true provided that $f(x)$ is bounded below.