Final answer:
A removable discontinuity occurs at a point where a function is not defined or not continuous, but can be redefined to be continuous. Continuous functions, such as horizontal lines, do not have these discontinuities. In probability, a continuous probability function represents the probability through the area under its curve.
Step-by-step explanation:
A removable discontinuity occurs when a function f (x) is either not defined at x = a, or not continuous at that point, but can be made continuous by redefining the function value at x = a. For instance, if the function f(x) = (x^2 - 1)/(x - 1) can be simplified to f(x) = x + 1 except at x = 1 where the function was not defined originally, then by defining f(1) = 2 (which is the limit of f(x) as x approaches 1), the discontinuity at x = 1 can be 'removed'. This contrasts with a continuous function, which does not have any discontinuities over its domain. An example of a continuous function is a horizontal line f(x) = k, for any constant k, over a closed interval such as [0, 20]. In probability theory, a continuous probability function is defined so that the area below the function and above the x-axis represents a probability, with the total area constrained to one since it corresponds to the maximum probability.