Discrete Fourier Transform of PMF and PDF
A random variable (RV) doesn't need to be continuous in order to have a PDF. The PDF of a discrete RV, i.e. a RV that takes on only discrete values, is of course not a continuous function but is only different from zero at the possible discrete values of the RV.

An example would be the outcome of a fair die. It takes on the values 1 to 6 with equal probability $1/6$. It's PDF is given by

$$p(x)=\frac16\sum_{i=1}^6\delta(x-i)\tag{1}$$

where $\delta(x)$ is the Dirac delta impulse. The Dirac delta impulses are necessary in the PDF (as opposed to the PMF) because probabilities are computed from the PDF by integration. Since for any interval $[a,b]$ containing the value $0$

$$\int_{a}^b\delta(x)dx=1$$

the PDF given by (1) integrates to $1$, as desired. And the probability of the RV $X$ being inside an arbitrary interval $[a,b]$ is

$$P[a\le X\le b]=\int_a^b p(x)dx$$