Lipschitz Continuity

  • All my life as an artist I have asked myself: What pushes me continually to make sculpture? I have found the answer. art is an action against death. It is a denial of death…Imagination is a very precise thing, you know-it is not fantasy; the man who invented the wheel while he was observing another man walking-that is imagination!… I am the most curious of all to see what will be the next thing that I will do. (Jacques Lipchitz)

The Cubist sculptor Jacques Lipchitz (1891-1973) was not at all related to the German mathematician Rudolf Lipschitz (1832-1903), who is remembered for requiring the function f(x) in the initial value problem IVP

  • \dot x(t)=f(x(t))\quad\mbox{for }t>0,\, x(0)=x^0,

where \dot x = \frac{dx}{dt}, to be Lipschitz continuous in order to guarantee that a unique solution exists.

The Lipchitz sculpture Happiness of Living is Lipschitz continuous.

Definition:  A function u:Q\rightarrow Q is said to be Lipschitz continuous if there is a constant L, called the Lipschitz constant, such that

  • \vert u(t+dt)-u(t)\vert\le L*\vert dt\vert\quad\mbox{for all } t,\, dt\in Q,

where we here allow dt to also be negative.

For a Lipschitz continuous function u(t) the difference du=u(t+dt)-u(t) is small if \vert dt\vert is small, up to the constant L, in the sense that

  • \vert du\vert \le L*\vert dt\vert.

We may say that a Lipschitz continuous function u(t) is locally close to a constant value in the sense that u(t+dt) deviates from u(t) less than L*\vert dt\vert. By this we don’t mean that u(t) is close to a constant over its entire span, just locally.

One can relax the concept of Lipschitz continuity to Hölder continuity requiring instead for some fixed constant 0<\alpha <1

  • \vert u(t+dt)-u(t)\vert\le L*\vert dt\vert^\alpha \quad\mbox{for all } t,\, dt.

Hölder continuity also expresses local constancy, but with a different measure.

In order for a function value u(t) to be well defined for a given argument t, it is necessary that the function u(s) is (Lipschitz or Hölder) continuous at s=t. This is because if t is an irrational number (not a rational number), then t it is not known exactly to all its decimals, and thus u(t) has to be replaced by the value u(t+dt) with t+dt a finite decimal approximation of t, and in order for the replacement to make sense we must be able to guarantee that u(t+dt) is close to u(t) if dt is small.

Extension of a Lipschitz Continous Function (optional)

The previous argument can be used to show that a Lipschitz continuous function $latex f:Q\rightarrow Q$ can uniquely be extended to a Lipschitz function f:R\rightarrow R. Can you write down the argument?

Hint: Is it sufficient to show that u(t+dt) is close to u(t) if dt is small?

The concept of Lipschitz continuity is naturally extended to a function u:I\rightarrow R defined on some interval or union of intervals I: A function u:I\rightarrow R is Lipschitz continuous with Lipschitz constant L if

  • \vert u(t+dt)-u(t)\vert\le L*\vert dt\vert\quad\mbox{for }t, t+dt\in Q ,

Two Lipschitz continuos functions defined on two adjoining intervals [a,b] and [b,c] can be definedas one function defined on the union of intervals [a,c] with possibly a jump discontinuity at thecommon point b. We can thus speak about such piecewise Lipschitz continuous functions, but not about discontinuos functions in general.

Extension to a Function u(x)

The concept of Lipschitz continuity is naturally extended to a function u(x) where x is a space variable, or some other variable, as follows:

Definition: A function u:\Omega\rightarrow R where \Omega is a domain in R^d, is said to be Lipschitz continuous with Lipschitz constant L, if

  • \vert u(x+dx)-u(x)\vert\le L*\vert dx\vert\quad\mbox{for all }x,\, x+dx\in \Omega .

As above we understand that a Lipschitz continuous function u:Q^d\rightarrow Q^d can uniquely be extended to Lipschitz continuous function u: R^d\rightarrow R^d.

A Horrible Function which is a Non-Function

In math books you can find the following specification of values x(t)

  • x(t)=1\quad\mbox{if }t\quad\mbox{is rational}
  • x(t)=0\quad\mbox{if }t\quad\mbox{is irrational}.

Obviously, this is not a Lipschitz or Hölder continuous function, and in fact it is not a function at all,because you can not in general tell if a given argument $t$ has a finite/periodic decimal expansion or not.

To Think About

  • Is the sum of two Lipschitz continuous functions, Lipschitz continuous?
  • What about other combinations of functions?
Two linear functions with different slope and Lipschitz constant.

A Lipschitz continuous function does not change more rapidly than a linear function.

Qualitative Definition of Continuity

The concept of Lipschitz continuity is a quantitative concept of continuity, with theLipschitz constant giving quantitative control of a difference in function value du in terms of difference in argument dt say, as \vert du\vert\le L\vert dt\vert, which in quantitative form expresses that

  • \vert du\vert is small if \vert dt\vert is small.

In most standard Calculus books such a purely qualitative concept of continuity is used, which has advantage of being “more general” but at the cost of being “less precise”, since it does not connect smallness of dt to smallness of du in any quantitative form.

Hölder continuity is also quantitative while allowing almost any generality.

The reason we use Lipschitz/Hölder continuity is that it is more precise than a purely qualitative concept and therefore easier to both understand and use, while the loss of generality does not have any real cost.

To Think About

  • Can you give an example of a function which is not Lipschitz continuous?
  • What about u(x)=\frac{1}{x} defined for 0<x\le 1?
                                                                        Leibniz’ desktop computer.

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