Editing Piecewise function (section)

== Continuity and differentiability of piecewise-defined functions ==

[[Image:Upper semi.svg|thumb|280px|Plot of the piecewise-[[quadratic function]] <math>f(x) = \left\{ \begin{array}{lll} x^2 & \text{if} & x < 0.707 \\ 1.5 - (x - 1.414)^2 & \text{if} & 0.707 \leq x \\ \end{array} \right.</math> Its only discontinuity is at <math>x_0 = 0.707</math>.]]
A piecewise-defined function is [[Continuous function|continuous]] on a given interval in its domain if the following conditions are met:
* its sub-functions are continuous on the corresponding intervals (subdomains),
* there is no discontinuity at an endpoint of any subdomain within that interval.

The pictured function, for example, is piecewise-continuous throughout its subdomains, but is not continuous on the entire domain, as it contains a jump discontinuity at <math>x_0</math>. The filled circle indicates that the value of the right sub-function is used in this position.

For a piecewise-defined function to be differentiable on a given interval in its domain, the following conditions have to fulfilled in addition to those for continuity above:
* its sub-functions are differentiable on the corresponding ''open'' intervals,
* the one-sided derivatives exist at all intervals' endpoints,
* at the points where two subintervals touch, the corresponding one-sided derivatives of the two neighboring subintervals coincide.<ref>{{SpringerEOM|title=One-sided derivative|author-last1=Rehmann|author-first1=Ulf|oldid=48044}}</ref><ref>{{Cite book |last=Ilyin |first=V. A. |title=Fundamentals Of Mathematical Analysis |last2=Poznyak |first2=E. G. |publisher=Mir Publishers Moscow |others=Translated from Russian by Irene Aleksanova. |year=1982 |isbn=978-93-859-2386-9 |volume=1 |pages=146, 177}}</ref><ref>{{Cite book |last=Canuto |first=Claudio |title=Mathematical Analysis I |last2=Tabacco |first2=Anita |publisher=Springer-Verlag Italia |others=Translated by: Simon G. Chiossi |year=2008 |isbn=978-88-470-0875-5 |location=Milan |publication-date=2008 |pages=83, 176}}</ref>