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Lipschitz continuity
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<p>In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, Lipschitz continuity, named after <a href="/facts/Germany/paluuWBX">German</a> <a href="/facts/Mathematician/eaFSR8Fx">mathematician</a> <a href="/facts/Rudolf_Lipschitz/g44HKdmH">Rudolf Lipschitz</a>, is a strong form of <a href="/facts/Uniform_continuity/hjvw7tII">uniform continuity</a> for <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>. Intuitively, a Lipschitz <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the <i>Lipschitz constant</i> of the function (and is related to the <i><a href="/facts/Modulus_of_continuity/31lThXhW">modulus of uniform continuity</a></i>). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. </p><p>In the theory of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>, Lipschitz continuity is the central condition of the <a href="/facts/Picard%E2%80%93Lindel%C3%B6f_theorem/yKHSWoMJ">Picard–Lindelöf theorem</a> which guarantees the existence and uniqueness of the solution to an <a href="/facts/Initial_value_problem/PqmJfGML">initial value problem</a>. A special type of Lipschitz continuity, called <a href="/facts/Contraction_mapping/VSu4qNqa">contraction</a>, is used in the <a href="/facts/Banach_fixed-point_theorem/TRygeoCP">Banach fixed-point theorem</a>. </p><p>We have the following chain of strict inclusions for functions over a <a href="/facts/Compactness/d0cgXJH7">closed and bounded</a> non-trivial interval of the real line: </p> <a href="/facts/Continuously_differentiable/jQqTgLk1">Continuously differentiable</a> ⊂ Lipschitz continuous ⊂ α {\displaystyle \alpha } -<a href="/facts/H%C3%B6lder_continuous/CI0yp6kD">Hölder continuous</a>, <p>where 0 < α ≤ 1 {\displaystyle 0<\alpha \leq 1} . We also have </p> Lipschitz continuous ⊂ <a href="/facts/Absolutely_continuous/NPKDt9Qn">absolutely continuous</a> ⊂ <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a> ⊂ <a href="/facts/Continuous_function/sKbl02pB">continuous</a>.