Can a random variable with a non-continuous cumulative density function have a probability density function?
Yes, this is possible, for example when we are dealing with random variables that are discrete in nature.
Indeed it is possible that a random variable with a non-continuous cumulative density function has a probability density function. A simple example is when we are dealing with discrete random variables. The following graph shows three different situations.
"From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part." - taken from figure caption at this link