Does Wolfram Mathworld make a mistake describing a discrete probability distribution with a probability density function? Usually a probability distribution over discrete variables is described using a probability mass function (PMF):

When working with continuous random variables, we describe probability distributions using a probability density function (PDF) rather than a probability mass function.
-- Deep Learning by Goodfellow, Bengio,  and Courville

However, Wolfram Mathworld is using PDF to describe the probability distribution over discrete variables:



Is this a mistake? or it does not much matter?
 A: In addition to the more theoretical answer in terms of measure theory, it is also convenient to not distinguish between pmfs and pdfs in statistical programming. For example, R has a wealth of built-in distributions. For each distribution, it has 4 functions. For example, for the normal distribution (from the help file):
dnorm gives the density, pnorm gives the distribution function, qnorm gives the quantile function, and rnorm generates random deviates.

R users rapidly become used to the d,p,q,r prefixes. It would be annoying if you had to do something like drop d and use m for e.g. the binomial distribution. Instead, everything is as an R user would expect:
dbinom gives the density, pbinom gives the distribution function, qbinom gives the quantile function and rbinom generates random deviates.

A: It is not a mistake
In the formal treatment of probability, via measure theory, a probability density function is a derivative of the probability measure of interest, taken with respect to a "dominating measure" (also called a "reference measure").  For discrete distributions over the integers, the probability mass function is a density function with respect to counting measure.  Since a probability mass function is a particular type of probability density function, you will sometimes find references like this that refer to it as a density function, and they are not wrong to refer to it this way.
In ordinary discourse on probability and statistics, one often avoids this terminology, and draws a distinction between "mass functions" (for discrete random variables) and "density functions" (for continuous random variables), in order to distinguish discrete and continuous distributions.  In other contexts, where one is stating holistic aspects of probability, it is often better to ignore the distinction and refer to both as "density functions".
