Consider the following probability mass function $f$:
$ f(x)= \begin{cases} 0&\text{}\, x\lt 0\\ \frac{x}{4}&\text{}\, 0\leq x\lt 1\\ \frac{1}{2}+\frac{x-1}{4}&\text{}\, 1\leq x\lt 2\\ \frac{11}{12}&\text{}\, 2\leq x\lt 3\\ 1&\text{}\, 3\leq x\\ \end{cases} $
Lets say we want to find $P(X=i)$, for $i=1,2,3.$
For $P(X=1)$, we can calculate it as $P(X\leq1)-P(X<1)$, which in this case would be $f(1)-f(1^{-})$.
=$(\frac{1}{2}+\frac{1-1}{4}) - \frac{1}{4}$
=$\frac{1}{4}$
I understand the concept of $f(1)-f(1^{-})$, but why are we only considering $f(1^{-})$, and not $f(1^{+})$ in this case?