Is it justifiable to call the probability mass function by the name “discrete probability density function”? Commonly, the probability density function (PDF) is used when dealing with continuous random variables, while the probability mass function (PMF) is used for discrete random variables.
This is the reason they are called “density function” and “mass function” respectively.
However, my professor would talk about a “continuous PDF” and a “discrete PDF”, instead of a PDF and PMF.
It seems that my professor is not the only one to use the term “discrete probability density function”. It is also used in these UBC lecture notes.
Is it correct to call the probability mass function by the name “discrete probability density function”?

Let $X$ be a discrete random variable with probability mass function $p_X : \mathcal{X} \to [0,1]$, where $\mathcal{X}$ is a discrete set (possibly countably infinite). Random variable $X$ can be thought of as a continuous random variable with the following probability density function
$$f_X (x) = \sum_{x_k \in \mathcal{X}} p_X  (x_k) \, \delta (x - x_k)$$
where $\delta$ is the Dirac delta. One could call $f_X$ a discrete probability density function, as its support is a discrete set. However, do note that $f_X$ is a generalized function, not a proper function.
