Limit Superior and Limit Inferior - Real-valued Functions

Real-valued Functions

Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and -∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given f(x) = sin(1/x), we have lim sup_x→0 f(x) = 1 and lim inf_x→0 f(x) = -1. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at a. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero . Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.