Final answer:
A function is considered discontinuous at values of x where there are breaks, holes, jumps, or undefined points like vertical asymptotes in the function's graph. To find these points, analyze where the function's formula changes, where division by zero or other undefined expressions occur, and examine the continuity of the function and its first derivative.
Step-by-step explanation:
To determine on what values of x a given function discontinuous, we need to analyze the function's behavior. A function is continuous if it does not have any breaks, holes, jumps, or asymptotes. Discontinuities can occur when the function is not defined at some point, if there is a vertical asymptote or if the left and right limits do not match at a certain point.
For instance, for the function y = 1/x, there is a discontinuity at x = 0 because the function approaches infinity as x approaches zero, and thus it is not defined at x = 0. In the context of the provided reference, a function must be both continuous and its first derivative must be continuous, unless a potential function V(x) = ∞, indicating a vertical asymptote or infinite potential barrier where the continuity breaks.
When assessing a function for discontinuities, we look at the points where the function's formula changes, where division by zero might occur, or where terms in the function may lead to undefined values. Each of these can indicate a discontinuity. To analyze a function comprehensively, we should also consider its domain, and the behavior of the function and its first derivative within that domain.