Final answer:
A continuous function must not exhibit any interruptions, and its first derivative must also be continuous, except where the potential is infinite. In probability, a continuous probability density function is used to represent probability as an area under the curve, which must be a smooth curve on the graph over its domain.
Step-by-step explanation:
A student asked if a function is continuous at all points on the disk. To determine if a function is continuous, it has to meet certain criteria. Specifically, the function y(x) must not exhibit any breaks, jumps, or points of discontinuity within the domain. A continuous function must also have a continuous first derivative, dy(x)/dx, except where the potential function V(x) is infinite. In the context of continuous probability functions, we consider the probability density function (pdf), denoted f(x), which describes the probability as an area under a curve on a graph; the total area under the curve is constrained to a maximum of one, corresponding to the maximum probability.
Regarding properties of a continuous probability distribution, the function f(x) which represents a probability density must be a smooth curve over its domain, with probability given by the area under this curve. For example, the probability P(x > 15) for a continuous probability distribution where 0≤x≤ 15 is zero, as x cannot exceed 15 within this domain. Similarly, the probability P(x = 7) where 0≤x≤ 10 would be zero for a continuous probability distribution, since the probability of a specific value in a continuous distribution is always zero.