Editing Differentiable function (section)

==Differentiability and continuity==
{{See also|Continuous function}}
[[File:Absolute value.svg|left|thumb|The [[absolute value]] function is continuous (i.e. it has no gaps). It is differentiable everywhere ''except'' at the point {{math|''x''}} = 0, where it makes a sharp turn as it crosses the {{math|''y''}}-axis.]]
[[File:Cusp at (0,0.5).svg|thumb|right|A [[cusp (singularity)|cusp]] on the graph of a continuous function. At zero, the function is continuous but not differentiable.]]
If {{math|''f''}} is differentiable at a point {{math|''x''<sub>0</sub>}}, then {{math|''f''}} must also be [[continuous function|continuous]] at {{math|''x''<sub>0</sub>}}.  In particular, any differentiable function must be continuous at every point in its domain. ''The converse does not hold'': a continuous function need not be differentiable. For example, a function with a bend, [[cusp (singularity)|cusp]], or [[vertical tangent]] may be continuous, but fails to be differentiable at the location of the anomaly.

Most functions that occur in practice have derivatives at all points or at [[Almost everywhere|almost every]] point. However, a result of [[Stefan Banach]] states that the set of functions that have a derivative at some point is a [[meagre set]] in the space of all continuous functions.<ref>{{cite journal |last=Banach |first=S. |title=Über die Baire'sche Kategorie gewisser Funktionenmengen |journal=[[Studia Mathematica|Studia Math.]] |volume=3 |issue=1 |year=1931 |pages=174–179 |doi=10.4064/sm-3-1-174-179 |doi-access=free }}.  Cited by {{cite book|author1=Hewitt, E  |author2=Stromberg, K|title=Real and abstract analysis|publisher=Springer-Verlag|year=1963|pages=Theorem 17.8|no-pp=true}}</ref> Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the [[Weierstrass function]].
{{-}}