Editing Lipschitz continuity (section)

==Examples==
;Lipschitz continuous functions that are everywhere differentiable:{{unordered list
 | The function <math>f(x)=\sqrt{x^2+5}</math> defined for all real numbers is Lipschitz continuous with the Lipschitz constant ''K''&nbsp;{{=}}&nbsp;1, because it is everywhere [[Differentiable function|differentiable]] and the absolute value of the derivative is bounded above by 1. See the first property listed below under "[[Lipschitz continuity#Properties|Properties]]".
 | Likewise, the [[sine]] function is Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.
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;Lipschitz continuous functions that are not everywhere differentiable:{{unordered list
 |The function <math>f(x) = |x|</math> defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the [[reverse triangle inequality]]. More generally, a [[norm (mathematics)|norm]] on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.}}
;Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable:{{unordered list
 | The function <math>f(x) \;=\; \begin{cases} x^2\sin (1/x) & \text{if }x \ne 0 \\ 0 & \text{if }x=0\end{cases}</math>, whose derivative exists but has an essential discontinuity at <math>x=0</math>.
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;Continuous functions that are not (globally) Lipschitz continuous:{{unordered list
 | The function ''f''(''x'')&nbsp;{{=}}&nbsp;{{radic|''x''}} defined on [0,&nbsp;1] is ''not'' Lipschitz continuous. This function becomes infinitely steep as ''x'' approaches 0 since its derivative becomes infinite. However, it is uniformly continuous,<ref>{{Citation | last1=Robbin | first1=Joel W. | title=Continuity and Uniform Continuity | url=http://www.math.wisc.edu/~robbin/521dir/cont.pdf}}</ref> and both [[Hölder continuity|Hölder continuous]] of class ''C''<sup>0,  α</sup> for α&nbsp;≤&nbsp;1/2 and also [[absolutely continuous]] on [0,&nbsp;1] (both of which imply the former).
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;Differentiable functions that are not (locally) Lipschitz continuous:{{unordered list
 | The function ''f'' defined by ''f''(0)&nbsp;{{=}}&nbsp;0 and ''f''(''x'')&nbsp;{{=}}&nbsp;''x''<sup>3/2</sup>sin(1/''x'') for 0<''x''≤1 gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.
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;Analytic functions that are not (globally) Lipschitz continuous:{{unordered list
 | The [[exponential function]] becomes arbitrarily steep as ''x'' → ∞, and therefore is ''not'' globally Lipschitz continuous, despite being an [[analytic function]].
 | The function ''f''(''x'')&nbsp;{{=}}&nbsp;''x''<sup>2</sup> with domain all real numbers is ''not'' Lipschitz continuous. This function becomes arbitrarily steep as ''x'' approaches infinity. It is however locally Lipschitz continuous.
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