Monotone functions — limits, jumps and integrability

Let $f:[0,1]\to[0,1]$ be a monotonically increasing function, that is, $a\le b$ implies that $f(a)\le f(b)$. For any $\alpha\in(0,1)$, let $L_{\alpha}^{+}=\lim_{x\to\alpha^{+}}f(x)$ and $L_{\alpha}^{-}=\lim_{x\to\alpha^{-}}f(x)$ denote the right hand and the left hand limits respectively, provided they exist. For $\alpha\in(0,1)$, if $L_{\alpha}^{+}$ and $L_{\alpha}^{-}$ exist, then define $U_{\alpha}=(L_{\alpha}^{-},L_{\alpha}^{+})$ if $L_{\alpha}^{-}<L_{\alpha}^{+}$, and $U_{\alpha}=\emptyset$ if $L_{\alpha}^{+}\le L_{\alpha}^{-}$. Which of the following statements are true?