is there a way how to get quantile function from random variable $Y$ defined as: $$ Y = \begin{cases} 0 \text{ ... with probability 0.25}\\ f(x) \text{ ... with probability 0.75} \end{cases} $$ where $f(x)$ is continuous probability distribution function?
Thank you for your time.
Let $F_Y^{-1}$ denote the quantile function of $Y$. Then $$ \forall u\in(0,1),\quad F_Y^{-1}(u)=\left\{\begin{array}{rcl}0&\textrm{if}&u\le\frac14\\f(x)&\textrm{if}&u>\frac14\end{array}\right. $$
If you want the details: Let $F_Y$ denote the cumulative distribution function of $Y$. Then $F_Y(y)=0$ if $y<0$, $F_Y(y)=\frac14$ if $0\le y<f(x)$ and $F_Y(y)=1$ if $y\ge f(x)$. You can then deduce the quantile function from its definition: $$ F_Y^{-1}(u)=\inf\{y\in\mathbb R\mid F_Y(y)\ge u\}. $$