Final answer:
A rational function is continuous if it has no points of discontinuity, typically where the denominator equals zero. For continuous probability distributions, probability is given by the area under the probability density function. Discontinuities, double-values, or divergence in the function can affect continuity and probabilities.
Step-by-step explanation:
To determine if a rational function is continuous, we need to check if the function doesn't have any points of discontinuity, which typically occur where the denominator is zero and cannot be canceled out by the numerator. In general, a rational function is continuous wherever it is defined, that is, for all real numbers except where the function has a denominator that equals zero. If the first derivative of the function with respect to space, dy(x)/dx, must also be continuous unless the function's value tends to infinity (denoted as V(x) = ∞).
When dealing with continuous probability distributions, the probability density function (pdf) f(x) is utilized, and probability is equated with the area below the curve of the graph. For continuous variables, such as money spent or received, probability is represented by area, whereas for discrete variables, such as the number of books bought or sold, probability is represented by individual counts. It's important to note that for a continuous probability distribution, probabilities of exact values, like P(x = 7), are zero since they represent a single point on a continuum.
In probability problems, if you have a range such as 0 ≤ x ≤ 15, the probability of x being greater than 15, P(x > 15), would be zero because the function does not include values greater than 15. Similarly, P(x < 0) would also be zero if the function is restricted to 0 ≤ x ≤ 5. If the function has a discontinuity, if it is double-valued, or if it diverges and is not normalizable, these instances can also affect the continuity and the way probabilities are calculated within a given range.