Final answer:
The function f(x) is continuous for all x within 0 ≤ x ≤ 20 since it is depicted as a horizontal line within the interval. No specific value of a is needed for continuity. For continuous probability distributions, we calculate probabilities over ranges, and the probability for an exact value is zero.
Step-by-step explanation:
The question asks about finding a value for a such that the function f(x) is continuous on the interval 0 ≤ x ≤ 20. Given that f(x) is represented as a horizontal line within this interval, continuity is inherent in the nature of a horizontal line, which does not break or change direction abruptly. Therefore, the function f(x) is already continuous for all values of x within the given interval, and there is no need to find a specific value of a to make it continuous.
When discussing continuous probability distributions, it's important to note that the probability of a single exact value, such as P(x = 7), is always 0. Instead, probabilities are given for ranges or intervals, like P(0 < x < 12), which represents the entire probability space for a continuous probability function restricted to 0 ≤ x ≤ 12.
For the question regarding the probability density function (PDF), the area under f(x) within the interval 0 ≤ x ≤ 20 is effectively the total probability and would be equal to 1, since a PDF must integrate to 1 over the entire space where it is defined.