Examples of Discontinuous Functions

Abstract criteria need a reference point. This chapter uses concrete examples to show what each kind of discontinuity looks like and what causes it, so that labels like “first kind” and “second kind” stop being bare terminology. Piecewise functions are the most typical sample: at a breakpoint the expressions on the two sides differ, so the function value either matches the one-sided limits (giving a removable discontinuity) or does not, and if the limit does not exist at all the discontinuity is of the second kind. The discontinuities of rational, trigonometric, exponential and logarithmic functions mostly come from restrictions in the domain — $\log x$ is simply undefined for $x \le 0$. Examining the common function families one by one is meant to build a habit: look at the domain and the critical points first, then verify point by point. That is more reliable than reciting the definition.