In order to understand the concept of the distribution function, it is essential to understand the concept of a continuous function. So, what do we mean when we refer to a continuous function? Basically, a continuous function is any function that does not have any unexpected changes in value. These abrupt or unexpected changes are referred to as discontinuities. For example, consider the following cubic function:

If you plot the graph of this function, you will see that there are no jumps or holes in the series of values. Hence, this function is continuous:

Having understood the continuous function, let's now try to understand what the probability density function (PDF) is. The PDF can be defined in terms of a continuous function, in other words, for any continuous function, the PDF is the probability that the variate has the value of x. 

Now, if you have been paying attention, an obvious question should pop up in your mind. What if the function is associated with discrete random variables rather than continuous random variables? Well, then the function is referred to as a probability mass function (PMF). For a more formal definition, refer to reference [6] in the Further reading section. 

The probability distribution or probability function of a discrete random variable is a list of probabilities linked to each of its attainable values. Let's assume that a random variable, A, takes all values over an interval of real numbers. Then, the probability that A is in the list of outcomes Z, P(Z), is the area above Z and under a curve that describes a function p(a) satisfying the following conditions:

1. The curve cannot have negative values (p(a) > 0 for all a).
2. The total area under the curve is always equal to 1.

Such curves are referred to as density curves. Continuous probability distributions include normal distribution, exponential distribution, uniform distribution, gamma distribution, Poisson distribution, and binomial distribution.