Final answer:
In probability, a function is continuous over its domain if it has no interruptions in its graph, allowing the area under the probability density function's curve to correctly represent probabilities, which must total 1 over the entire domain.
Step-by-step explanation:
A function is continuous at every number in its domain if there are no breaks, jumps, or holes in its graph within that domain. In the context of a continuous probability density function (pdf), this means the function f(x) is smooth and unbroken over its entire range. This continuity is essential because, in probability theory, we define probability as the area under the curve of the pdf, which must be a complete and uninterrupted shape to represent probabilities accurately. An important example of such a continuous function is the uniform probability distribution, where the probability between any two points is proportional to the length of the interval between them.
The total area under the curve of a probability density function must equal 1, as this represents the certainty of the entire sample space. For any continuous random variable X, P(x = c) = 0 because a single point has no width and thus no area under the curve. Instead, we look at intervals, such as P(a < x < b), to calculate probabilities for continuous distributions.
For the example given, f(x) defined for 0 ≤ x ≤ 20 being a horizontal line and therefore having a continuous first derivative concerning x, fulfills the criteria of being a continuous function in its domain. As for P(x = c), the probability is zero for any particular value because we deal with ranges in continuous distributions.