Continuous function
Definition using oscillation

Continuity can also be defined in terms of oscillation: a function f is continuous at a point x 0 {\displaystyle x_{0}} if and only if its oscillation at that point is zero; in symbols, ω f (x 0) = 0. {\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε {\displaystyle \varepsilon } (hence a G δ {\displaystyle G_{\delta }} set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.

The oscillation is equivalent to the ε − δ {\displaystyle \varepsilon -\delta } definition by a simple re-arrangement and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ε 0 {\displaystyle \varepsilon _{0}} there is no δ {\displaystyle \delta } that satisfies the ε − δ {\displaystyle \varepsilon -\delta } definition, then the oscillation is at least ε 0, {\displaystyle \varepsilon _{0},} and conversely if for every ε {\displaystyle \varepsilon } there is a desired δ, {\displaystyle \delta,} the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
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