8 Left-continuous inverses
Let \(R\) and \(S\) be complete linear orders, for example \([0,1]\), \([0,+\infty ]\), or \([-\infty ,+\infty ] =: \overline{\mathbb {R}}\).
Let \(f \colon R \to S\) be a function (usually assumed nondecreasing). The left-continuous inverse of \(f\) is the function \({f}^{\rightarrow 1} \colon S \to R\) given by
\begin{align*} {f}^{\rightarrow 1}(y) := \inf \big\{ x \in R \; \big| \; f(x) \ge y \big\} \qquad \text{ for } y \in S. \end{align*}
The right-continuous inverse \({f}^{\leftarrow 1} \colon S \to R\) is analoguously defined by
\begin{align*} {f}^{\leftarrow 1}(y) := \sup \big\{ x \in R \; \big| \; f(x) \le y \big\} \qquad \text{ for } y \in S. \end{align*}