Right Continuous Inverse

Last updated Nov 1, 2022

Let $F: \mathbb{R} \to \mathbb{R}$ be a Distribution Function. The Right Continuous Inverse of $F$ is defined

$$F^{\rightarrow}(x) = \inf {s \in \mathbb{R} : F(s) > x}$$

1. $F^{\rightarrow}$ is Non-Decreasing Function.

   Proof: Let $A_{x} = {s \in \mathbb{R} : F(s) > x}$. Suppose $y \leq x$. Then for all $s \in A_{x}$, we have that $s > x \geq y$ so $s \in A_{y}$. Thus, $A_{x} \subset A_{y}$. Since Infimums are Non-Increasing Set Functions, we have that $F^{\rightarrow}(y) \leq F^{\rightarrow}(x)$. $\blacksquare$

2. $F^{\rightarrow}$ is Right-Continuous.

   Proof: TODO

3. $\lim\limits_{t \uparrow x} F^{\rightarrow}(t) = F^{\leftarrow}(x)$

   Proof: TODO

• Real Numbers
• Left Function Limit
• Set