Final answer:
A function is continuous at a point if it is defined at that point, the limit of the function as it approaches that point is equal to its value there, and for probability functions, the area under the curve is finite and non-negative. If a function has discontinuities, is double-valued, or diverges, it is not continuous at those points. For continuous probability distributions, the probability of a specific point is always 0, as it represents no area over the x-axis.
Step-by-step explanation:
You are asking about the continuity of a function in the context of continuous probability functions. A function, f(x), is considered continuous at a point r if it meets two conditions: the function must be defined at r, and the limit of the function as it approaches r must be equal to the function’s value at r. Additionally, for a function to be a continuous probability density function, the area under the curve, which represents probability, must be finite and non-negative within its defined range.
From the information provided, if the function has a discontinuity, is double-valued, or diverges (meaning it is not normalizable), it would not be considered continuous at those problematic points. Specifically, for continuous probability distributions:
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- If you are looking for P(x > 15) and the function is defined from 0 to 15, then P(x > 15) would be 0 because there is no area under the curve beyond x = 15.
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- If you are asked for P(x = 7) or P(x = 10), the probability of a specific point for a continuous distribution is 0, as a point does not cover any area over the x-axis.
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- For P(x < 0), if the function is restricted from 0 to 5, then P(x < 0) would be 0 again because there is no area under the curve before x = 0.
Hence, the function f(x) is continuous at point r if there is no discontinuity, it is not double-valued at r, and it does not diverge at r. Within the range the function is defined and normalizable, these are the points where it will be continuous