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?

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 ${Q_{X}(0.2)=0}$.
