Final answer:
The point at which the function r² is continuous is determined by examining if the function is defined, finite, and without discontinuities over its domain. In probability theory, continuous probability density functions must be continuous to calculate probabilities as areas under the curve, adhering to the rule that the sum of all probabilities must equal one.
Step-by-step explanation:
Understanding Continuity in Probability Density Functions
The question posed is about the continuity of a function denoted as r² which seems to be related to continuous probability functions. In probability theory and statistics, a continuous probability density function (pdf) is a function that describes the likelihood of a continuous random variable taking on a particular value. The function must be continuous over its domain, meaning that there are no abrupt changes or gaps in its graph. If we want to determine at what points the function r² is continuous, we should look for any points where the function may not be defined, has infinite values, or has discontinuities, such as jumps or holes.
For continuous random variables, the probability of the variable taking on any specific single point value is zero, which is expressed as
. Instead, probabilities are defined for intervals and are calculated by finding the area under the probability density function, between two points. As per the provided information, f(x) is a pdf for a continuous random variable, hence the total area under this curve and above the x-axis must equal one, encapsulating the fact that the sum of all probabilities in a distribution is one.
Continuity of
is crucial because only with a continuous function can we properly calculate probabilities as areas under a curve. Discontinuities in
would result in undefined areas and hence undefined probabilities for certain intervals, which violate the basic properties of a probability density function.