Discrete probability, continuous density: in search of a unified and consistent nomenclature

Hello,

For discrete r.v.s we have the probability function, e.g., P(X = 5).

For continuous r.v.s. we have the density function, e.g., f(x).

But is there any way to treat discrete probability functions
and continuous density functions in a unified fashion?
That is, is there any "common word" for discrete probabilities
and continuous densities?

Obviously, such treatment would be massively helpful
when we don't wish to make assumptions
about the continuity of RVs.

So far I have got two ideas, but I'm not sure
whether they are statistically "meaningful"...

1.) Use the term "density function" for both discrete
and continuous r.v.s. There is some intuitive appeal
behind this idea, but rigorous treatment is probably
complicated, I would think.

2.) Use the term "likelihood function" for both discrete
and continuous r.v.s. Technically, a likelihood function
is defined as the density (or probability) of given
data with respect to an unknown parameter.
Therefore, this "trick" would neatly unify the
treatment of continuous and discrete r.v.s,
but at the cost of technical correctness, I would think.

I apologize if this sounds confusing, I wasn't originally
trained as a statistician. Any help or literature
pointers are greatly appreciated.

Thanks,
- Mikko
.

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