Monotonic functions are always riemann integrable jumps and all integrability says nothing about continuity

Let f:[0,1]→[0,1] be a monotonically increasing function, that is, a≤b implies f(a)≤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 left hand limits respectively, provided they exist. For $\alpha\in(0,1)$, if $L_\alpha$⁺ and $L_\alpha$⁻ exist, define $U_\alpha=(L_\alpha$⁻,$L_\alpha$⁺) if $L_\alpha$⁻$<L_\alpha$⁺, and ∅ if $L_\alpha$⁺$\le{}L_\alpha$⁻. Which of the following statements are true?