Final answer:
The statement is true; a function is continuous on an open interval (a,b) if it is continuous at each point on that interval, necessary for defining probabilities and ensuring the validity of probability density functions in continuous distributions.
Step-by-step explanation:
The statement that a function is continuous on an open interval (a,b) if it is continuous at each point on the interval is true. A continuous function means that there are no breaks, jumps, or holes in the graph of the function for any value within that interval. In other words, for the function y(x), the value of y(x) must be defined at every point and the first derivative of y(x) with respect to space, dy(x)/dx, should also be continuous unless the potential V(x) is infinite, as this would imply a discontinuity.
For example, in continuous probability functions, the probability density function (pdf) is used to model continuous random variables, where the probability is equal to the area under the pdf graph between two points. Thus, for the pdf to be valid, it must be continuous so that the area (and thus the probability) can be well-defined between any two points within the interval of interest. Similarly, when considering a uniform distribution, the function must remain continuous over its domain to ensure that all outcomes are equally likely.