# 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.

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.

• Is the sum of two Lipschitz continuous functions, Lipschitz continuous?
• What about other combinations of functions?

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.

• What about $u(x)=\frac{1}{x}$ defined for $0?