Functions of First Class

Abstract

Let f be a real-valued function on R. For any interval I, the quantity

Let f be a real-valued function on R. For any interval I, the quantity

$$w(I) = \mathop {\sup }\limits_{x \in I} f(x) - \mathop {\inf }\limits_{x \in I} f(x)$$

is called the oscillation of f on I. For any fixed x, the function ω((x–δ, x+δ)) descreases with δ and approaches a limit

$$\omega (x) = \mathop {\lim }\limits_{\delta \to 0} \;\omega ((x - \delta ,\;x + \delta )),\;$$

called the oscillation of f at x. ω(x) is an extended real-valued function on R. Evidently, ω(x₀)=0 if and only if f is continuous at x₀. When it is not zero, ω(x₀) is a measure of the size of the discontinuity of f at x₀.