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Near $(\bar{u},\bar{v})$ in a graph, this property requires neighborhoods $U,V$ and a finite $\kappa$ such that $S(u)\cap V\subset S(u^{\prime })+\kappa \Vert u-u^{\prime }\Vert \mathbb{B}$ for all $u,u^{\prime }\in U$. Nearby solutions can be matched under nearby perturbations with linear displacement control. Single-valued maps reduce to local Lipschitz continuity.

A causal curve has an everywhere nonspacelike tangent of consistent time orientation; its smooth segments are timelike or null. More generally, curves with local Lipschitz continuity and future causal tangent almost everywhere are allowed. The causal future and causal past are defined by reachability along these curves.

If $E\subseteq [0,1]$ is a Lebesgue measurable set and $0\le b\le \lambda (E)$, some measurable subset $B\subseteq E$ has Lebesgue measure $b$. The function $F(t)=\lambda (E\cap [0,t])$ has Lipschitz continuity with constant $1$, starts at $0$ and ends at $\lambda (E)$; the intermediate value theorem proves the assertion. Applying the assertion successively to remaining portions of $E$ gives pairwise disjoint measurable pieces with any finite list of nonnegative measures whose sum is at most $\lambda (E)$. This is a concrete consequence of non-atomic measure structure, which fails for general measures containing an atom of a measure.

Let $v\le 0$ be $C^1$ on a connected domain and $C^2$ away from its zero set, with $\Delta v+v=0$ wherever $v<0$. If negative and zero points coexist, an interior Euclidean ball in $\{v<0\}$ touches its zero set. On that Euclidean ball $\Delta v=-v>0$, so the Hopf boundary point lemma forces a nonzero derivative at an interior maximum, contrary to $Dv=0$. Thus $v$ is identically zero or strictly negative. Mere Lipschitz continuity does not suffice: $-\max \{\sin x_1,0\}$ on $(-1,1{)}^n$ has both negative and zero regions.

Set $\mu (0)=0$ and extend $\mu (r)=1$ for $r\ge 1$. This positive, nondecreasing modulus of continuity is weaker than a linear Lipschitz continuity bound but still has ${\int }_0^{1}dr/\mu (r)=\infty$. At small distances it is equivalent to $r\log (1/r)$. A signed $r\log r$ cannot be an upper modulus at distances less than one. The planar vorticity velocity kernel maps $L^1\cap L^{\infty }$ into velocity fields with this modulus.

For $d<p<\infty$, every Sobolev space element $u\in W^{1,p}({\mathbb{R}}^d)$ has a Hölder continuous function representative of exponent $1-d/p$, with the displayed bound. It also satisfies $\Vert u{\Vert }_{\infty }\le C_{d,p}(\Vert u{\Vert }_p+\Vert \nabla u{\Vert }_p)$. Averaging the fundamental theorem of calculus along a line segment over a ball givesThe Holder inequality bounds this by $C_{d,p}{r}^{1-d/p}\Vert \nabla u{\Vert }_p$, because $(d-1)p^{\prime }<d$. To compare two ball averages, translate a ball along the segment between their centres and use the same fundamental theorem of calculus along a line segment. These bounds give the displayed estimate. Density of smooth functions in a Sobolev space supplies the representative for nonsmooth $u$. For $p=\infty$, the corresponding conclusion is Lipschitz continuity.

A one-sided Lipschitz condition controls the growth of distances between solutions of an ordinary differential equation. Differentiating the squared difference and applying the Gronwall inequality gives $\Vert y(t)-z(t)\Vert \le e^{\nu t}\Vert y(0)-z(0)\Vert$. The case $\nu \le 0$ is a dissipative vector field. Ordinary Lipschitz continuity implies such a bound, but the converse fails: $f(y)=-y^3$ has one-sided constant zero on the real line without being globally Lipschitz continuous.

The empirical spectral measures of the two real symmetric matrices arewhere ${\delta }_t$ is the Dirac measure at $t$. Thus $⟨L_N,f⟩=N^{-1}{\sum }_{i}f({\lambda }_i)$. Pair the eigenvalues in the specified increasing order and apply the triangle inequality followed by Lipschitz continuity with bound $1$:This deterministic estimate uses only the Lipschitz bound; no entry independence or distributional hypothesis is needed here.

A function $f:\mathbb{R}\to \mathbb{R}$ satisfies Lipschitz continuity with bound $L\ge 0$ ifHere $L$ is an admissible Lipschitz bound; the least admissible value is the Lipschitz constant. The later parts use the admissible bound $1$.

The characteristic equations for a transport equation on phase space areA sufficient global hypothesis is that $F$ is continuous in time, locally Lipschitz continuous in $(x,v)$ uniformly on compact time intervals, and, for every finite $T$, satisfies $|F(t,x,v)|\le C_T(1+|x|+|v|)$ for $|t|\le T$. These hypotheses give unique characteristic curves for all real times if the field is defined for all real times. It suffices for forward existence to impose them on nonnegative time. One may strengthen the spatial hypothesis to $C^1$ when differentiating the characteristic flow map below. Local Lipschitz continuity alone guarantees only local existence; the linear-growth bound prevents finite-time escape.