Final answer:
If the integral of a function f(x) exists from b to t for every number t ≤ b, this implies that the function f(x) is Riemann integrable, which is necessary for functions to be considered continuous probability density functions in probability theory.the options provided in the student's question, the correct option is C: The function f(x) is Riemann integrable.
Step-by-step explanation:
If the integral of a function f(x) exists from b to t for every number t ≤ b, this suggests that the function has a certain property. It does not necessarily mean that the function is continuous, differentiable, or periodic, although these properties could be related to a function's integrability. What this does imply is that the function is Riemann integrable.
Riemann integrability is a criterion which essentially means that the function can be approximated by sums of products of function values and subinterval lengths, and that mapping this approximation as the subintervals shrink in size will lead to a definite value, which is the integral of the function over that interval. This characteristic allows for the calculation of the area under the curve of the function on a given interval, which is crucial in many fields, including probability theory.
For continuous probability density functions, the area under the curve between two points on the x-axis represents the probability that a random variable takes a value between those points. In this context, if f(x) is the continuous probability density function for a random variable, then the total area under the curve f(x) over the interval from a to b must be equal to the probability P(a < x < b), hence the integral must exist for the function to be a proper probability density function.
Returning to the options provided in the student's question, the correct option is C: The function f(x) is Riemann integrable.