Quantile function of CDF

Looking at the CDF, notice that we are actually dealing with a discrete random variable with 
\[p(0)=0.5,  p(1)=0.3,  \text{and}    p(2)=0.2.\]
The quantile function you have captures the essense of this CDF, but you can write other  quantile function by using values of $p$ other than $0.5$ and $0.8$. For example we can define
\[I(p)= \{0\}  \text{for}   p \in (0,0.6]  \text{since}  F(\{0\}) \subset (0,0.6] \]
\[I(p)= (0,1]  \text{for}   p \in (0.6,0.9]  \text{since}  F((0,1] ) \subset (0.6,0.9]\]
\[I(p)= \{2\}   \text{for}   p \in (0.9,1)  \text{since}  F(\{2\} ) \subset (0.9,1).\]

Note that if you define the quantile function $F^{-1}: (0,1)\rightarrow \mathbb{R}$ by
$$F^{-1}(a)=\inf \{x \in \mathbb{R}:F(x)\geq a\}.$$
Then the quantile function is unique, and is the one that you have obtained. But there are looser defiitions of quantile that allow for more than one quantile function. 

This took me over 30 minutes to figure out and answer. Please offere higher bounties in future, otherwise you may not get a response.