I was researching why it is that the cumulative distribution function of a discrete random variable could not exist at a median of $0.5$ in some cases. I came across this answer by the user V.Vancak, which answered my initial question. But at the end of his post he says
... hence you have to chose $0$ or $1$ as the median. Any value in $(0,1)$ will not change a thing as $X$ cannot receive these values.
If the cumulative distribution function does not exist at a median of $0.5$, then why, according to V.Vancak, must we select $0$ or $1$ as the new median? How does that solve the issue of there being no value of the CDF that will satisfy a median of $0.5$? Why will no other value in $(0, 1)$ suffice?
I would greatly appreciate it if people could please take the time to clarify this.