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 Gδ 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.
Read more about this topic: Oscillation (mathematics)
Famous quotes containing the word continuity:
“The dialectic between change and continuity is a painful but deeply instructive one, in personal life as in the life of a people. To “see the light” too often has meant rejecting the treasures found in darkness.”
—Adrienne Rich (b. 1929)
“There is never a beginning, there is never an end, to the inexplicable continuity of this web of God, but always circular power returning into itself.”
—Ralph Waldo Emerson (1803–1882)
“If you associate enough with older people who do enjoy their lives, who are not stored away in any golden ghettos, you will gain a sense of continuity and of the possibility for a full life.”
—Margaret Mead (1901–1978)