Generalized inverse of a function It is well-known that if a function is strictly increasing, then it has an inverse function. I also see the concept of "generalized inverse" in the litarature, which has the definition 
$$f^{-1}(x)=\inf\{y: f(y)>x\}.$$
What is the motivation of definition and can you give me examples which has not ordinary inverse but has generalized inverse?
 A: Generalized inverses of increasing functions are used in several domains such as function analysis, measure theory, probability and fuzzy logic. Several definitions of the generalized inverse are known leading to different properties. This paper aims at giving a precise study of the link between the definitions and the properties. It is shown why the right-continuous generalized inverse is a good choice. This rigorous treatment opens the doors to more focused studies.
Some references:
https://people.math.ethz.ch/~embrecht/ftp/generalized_inverse.pdf
https://hal-mines-paristech.archives-ouvertes.fr/hal-01255512/file/Report-GeneralizedInverse-partI.pdf
https://www.springerprofessional.de/en/the-generalized-inverse-of-distribution-functions/2332208
A: With reference to a continuous and strictly monotonic distribution function, for example the cumulative distribution function ${ F_{X}\colon R\to [0,1]}$ of a random variable ${X}$, the quantile function is defined by
\begin{equation*}
F_{X}(x):=\operatorname{Pr}(X \leq x)=p.   
\end{equation*}
Defining quantile functions for discrete rather than continuous distributions requires a bit more work since the discrete nature of such a distribution means that there may be gaps between values in the domain of the distribution function and/or "plateaus" in its range. Therefore, one often defines the associated quantile function (inverse distribution function)  to be
\begin{equation*}
 F^{-1}_{X}(x)=\inf \{x \in \mathbb{R}: p \leq F(x)\}. 
\end{equation*}
Example: Let ${X}$ be a discrete random variable with $R_{X}=\{0,1,2\}$ and probability mass function
\begin{equation*}
P_{X}(x)=\left\{\begin{array}{ll}
{1}/{2} & \text { if } x=0 \\
{1}/{4} & \text { if } x=1 \\
{1}/{4} & \text { if } x=2 \\
0 & \text { Otherwise }
\end{array}\right.
\end{equation*}
The distribution function of ${X}$ is
\begin{equation*}
F_{X}(x)=\left\{\begin{array}{cc}
0 & \text { if } x=0 \\
1 / 2 & \text { if } 0 \leq x<1 \\
3 / 4 & \text { if } 1<x<2 \\
1 & \text { if } x \geq 2
\end{array}\right.
\end{equation*}
Compute the ${p}$-quantile for ${p=0.2}$:
There is no such that ${F_{X}(x)=0.2}$.
However, the smallest ${x}$ such that ${F_{X}(x) \geq 0.2}$ is ${x=0}$ because ${F_{X}(x)=0}$ for ${x=0}$. Thus, we have ${Q_{X}(0.2)=0}$.
