As noted by Wikipedia, probability distribution function is ambiguous term:
A probability distribution function is some function that may be used
to define a particular probability distribution. Depending upon which
text is consulted, the term may refer to:
- a cumulative distribution function,
- a probability mass function, and/or
- a probability density function.
Cumulative distribution function (CDF) is sometimes shortened as "distribution function", it's
$$ F(x) = \Pr(X \le x) $$
the definition is the same for both discrete and continuous random variables. In dice case it's probability that the outcome of your roll will be $x$ or smaller.
Probability density function (PDF) is a continuous equivalent of discrete probability mass function (PMF). Probability mass function is
$$ f(x) = \Pr(X = x) $$
In dice case it's probability that the outcome of your roll will be exactly $x$.
Probability mass function has no sense for continuous random variables since $\Pr(X=x)=0$ for continuous random variables (check also Why X=x is impossible for continuous random variables?), because simply a point on real line is so "small" that has no mass and no area.
This leads us to defining probability density as "probability per foot". Simple example is continuous uniform distribution with minimum of $a$ and maximum of $b$, where probability density is the same for each $x$ and equal to
$$ f(x) = \frac{1}{b-a} $$
You can easily notice that it changes as the range between $a$ and $b$ (i.e the total area) changes, it is nicely described in Can a probability distribution value exceeding 1 be OK? thread. It is a probability of hitting infinitesimal (infinitely small) interval $[x, x + dx]$ when throwing a dice with infinite number of walls.
