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Is there any number a such that f(x) is continuous? If so, find all the possible value of such number a. If not, state your reason

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Only when x is equal to its restricted value
User Artalus
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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.

User Monkey Blot
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