Lipschitzian function
In mathematics, a function f: M → N between metric spaces (M, dM) and (N,dN) is said to be lipschitzian (either said to satisfy a Lipschitz condition or to be Lipschitz continuous) if there exists a constant K > 0 such that:
dN(f(x),f(and))≤ ≤ KdM(x,and),Русский Русский x,and한 한 M{displaystyle d_{N}(f(x),f(y))leq Kd_{M}(x,y), forall x,yin M}
In such a case, K is called the Lipschitz constant of the function. The name comes from the German mathematician Rudolf Lipschitz. For functions defined on Euclidean spaces, the previous relation can be written:
f(x)− − f(and) ≤ ≤ K x− − and ,Русский Русский x,and한 한 Rn,f:Rn→ → Rm{displaystyle intf(x)-f(y)informedleq Kindx-yind,forall x,yin mathbb {R} ^{n},qquad f:mathbb {R} ^{n}to mathbb {R} ^{m}}
Main features and results
- All Lipschitz function continues to be uniformly continuous and therefore continuous.
- Continuous Lipschitz features with constant Lipschitz K = 1 are called short functions and with K ≤1 receive the name of contractions. The latter are the ones that allow to apply Banach's fixed point theorem.
- The condition of Lipschitz is an important hypothesis to demonstrate the existence and uniqueness of solutions for ordinary differential equations. The condition of continuity of function alone ensures the existence of solutions (Peano Theorem), but in order to confirm also the uniqueness of the solution it is necessary to consider also the condition of Lipschitz (Picard-Lindelöf Theorem).
- Yeah. U is a subset of metric space M and f: U → R is a continuous Lipschitz function at real values, then there is always a continuous Lipschitz function M → R extending f and has the same Lipschitz constant as f(see also theorem of Kirszbraun).
- A continuous Lipschitz function f: I → R, where I is an interval in R, it is differential almost everywhere (always except in a set of measurements of Lebesgue zero). Besides, if K is the constant Lipschitzf, then. 日本語 (x) whenever the derivative exists. reciprocally, yes f: I → R is a differentiable function with embedded derivative, 日本語 (x) L for all x in I, then f is Lipschitz continuous with constant Lipschitz K ≥ Las a result of the theorem of the average value.
Related definitions
These definitions are required in the Picard-Lindelöf Theorem and related results.
- Locality Lipschitz: Givens M, N, metric spaces, a function is said f:MΔ Δ N{displaystyle f:Mlongrightarrow N} That's it. locally lipschitz if for every point of M there is an environment where the function fulfills the Lipschitz condition.
- Lipschitz function with a variable: Givens M, N, L metric spaces, a function is said f:M× × N→ → L(t,x) f(t,x){displaystyle {begin{matrix}f:Mtimes Nto L(t,x)mapsto f(t,x)end{matrix}}}}} That's it. locally Lipschitz on x{displaystyle x} if you meet the Lipschitz condition for N points.
