Editing Continuous function (section)

====Definition using oscillation====
[[File:Rapid Oscillation.svg|thumb|The failure of a function to be continuous at a point is quantified by its [[Oscillation (mathematics)|oscillation]].]]
Continuity can also be defined in terms of [[Oscillation (mathematics)|oscillation]]: a function ''f'' is continuous at a point <math>x_0</math> if and only if its oscillation at that point is zero;<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, Theorem 3.5.2, p. 172</ref> in symbols, <math>\omega_f(x_0) = 0.</math> A benefit of this definition is that it {{em|quantifies}} discontinuity: the oscillation gives how {{em|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 <math>\varepsilon</math> (hence a [[G-delta set|<math>G_{\delta}</math> set]]) – and gives a rapid proof of one direction of the [[Lebesgue integrability condition]].<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177</ref>

The oscillation is equivalent to the <math>\varepsilon-\delta</math> 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 <math>\varepsilon_0</math> there is no <math>\delta</math> that satisfies the <math>\varepsilon-\delta</math> definition, then the oscillation is at least <math>\varepsilon_0,</math> and conversely if for every <math>\varepsilon</math> there is a desired <math>\delta,</math> the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a [[metric space]].