Oscillation (mathematics) - Continuity

Continuity

Oscillation can be used to define continuity of a function, and is easily equivalent to the usual ε-δ definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point x₀ if and only if the oscillation is zero; in symbols, A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

For example, in the classification of discontinuities:

• in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
• in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
• in an essential discontinuity, oscillation measures the failure of a limit to exist.

This definition is useful 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 (hence a G_δ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.

The oscillation is equivalence to the ε-δ 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 ε₀ there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε₀, and conversely if for every ε there is a desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.