= Lipschitz continuity
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{title2=$d_Y(f(x),f(y))\leq Ld_X(x,y)$}
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= Lipschitz condition
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= Lipschitz continuous
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= Lipschitz function
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= Lipschitz
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= Lipschitz map
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A <function> $f:X\to Y$ between <metric spaces> is Lipschitz continuous if one finite number $L\geq0$ controls every change of output by the corresponding change of input:
$$
d_Y(f(x),f(y))\leq Ld_X(x,y)\qquad(x,y\in X).
$$
For real inputs and outputs this is $|f(x)-f(y)|\leq L|x-y|$. The same bound must work for every pair in the specified <domain of a function>. It gives a quantitative version of <continuity>: an input error of at most $\varepsilon$ produces an output error of at most $L\varepsilon$. The least possible bound is the <Lipschitz constant>. Lebl presents the real and metric-space definitions and their connection with <uniform continuity>. https://www.jirka.org/ra/realanal.pdf[Jiří Lebl, Basic Analysis I, Sections 3.4 and 7.5; accessed 2026-10-12]{ref}
Three <functions> show why this condition differs from <differentiability>. The <absolute value> $f(x)=|x|$ has a corner at zero, but the <reverse triangle inequality> gives $||x|-|y||\leq|x-y|$, so it has <Lipschitz constant> one on the <real line>. The <sine> is smooth and also has <Lipschitz constant> one: the <mean value theorem> and $|\cos t|\leq1$ bound every secant slope. The <square root> is <continuous> on $[0,1]$, but
$$
\frac{|\sqrt h-\sqrt0|}{|h-0|}=\frac1{\sqrt h}\longrightarrow\infty
\qquad(h\downarrow0).
$$
No finite <Lipschitz bound> works on that <interval>. At $h=1,1/4,1/16,1/64$, the ratios are $1,2,4,8$. Smaller input changes near zero therefore require larger amplification factors.
\Image[/lipschitz-slope-comparison.png]
{title=Bounded secant slopes permit a corner, while the <square root> needs unbounded amplification near zero}
For a real <function>, <Lipschitz continuity> means that the absolute slopes of all <secant lines> are bounded. If the <derivative> exists at a point, taking a <limit> of those slopes gives $|f'(x)|\leq L$. In the other direction, a <function> that is <continuous> on a real <interval>, <differentiable> in its interior, and satisfies $|f'|\leq L$ there is <Lipschitz continuous> by the <mean value theorem>. The <interval> hypothesis matters: the theorem uses the segment connecting the two inputs.
A <Lipschitz function> is <uniformly continuous>: for $L>0$, choosing input tolerance $\delta=\varepsilon/L$ works at every point. If $L=0$, the function is constant. The converse fails; the <square root> on $[0,1]$ is <uniformly continuous>, and satisfies the weaker <Hölder continuity> bound $|\sqrt x-\sqrt y|\leq\sqrt{|x-y|}$, without a finite <Lipschitz constant>. These distinctions are developed in Lebl's worked examples. https://www.jirka.org/ra/realanal.pdf[Lebl, Section 3.4, Proposition 3.4.8 and Examples 3.4.9–3.4.10]{ref}
The usefulness is controlled sensitivity. Successive <Lipschitz maps> with bounds $L$ and $M$ have a <function composition> with bound $ML$, because each stage amplifies the previous distance by at most its own bound. For an <ordinary differential equation>, <Lipschitz continuity> in the unknown state is the condition that makes nearby trial solutions controllably close in the <Picard-Lindelöf theorem>. For incomplete data, the <McShane extension theorem> fills in unobserved values while preserving the same bound. This one inequality connects error estimates, uniqueness, and <interpolation>.
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