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Lipschitz continuity is an even stronger form of uniform continuity and is often used in the theory of differential equations.

Derivation
By introducing the notion of Lipschitz continuity, we will be able to say more about the rate at which a function changes. As we already know, continuous functions have the neat property that sufficiently small changes in the arguments will result in arbitrarly small changes in the function value. On top of that, Lipschitz continuity allows us give an estimate of that change. We can thus talk very precisely about how "fast" the changes of the output become smaller. For better understanding, let us first recap what we mean by changes of a function. Let $$f:D\to\R$$ be an function with domain $$D$$.

Let's take two arbitrary points $$\tilde x$$ and $$x$$ from the domain of $$f$$ and form the straight line through the points $$(\tilde x, f(\tilde x))$$ und $$(x, f(x))$$. The slope of the line increases, as the difference between the values $$f(\tilde x)$$ and $$f(x)$$ gets bigger. The mean rate of change between function values is equal to the slope $\frac{f(x)-f(\tilde x)}{x-\tilde x} $ of the secant through those points and can be calculated with the so-called slope triangle:



Let's assume now that the rate of change is bounded, i.e. the slopes of the secants cannot get arbitrarly large or small. Thus the absolute value $\left|\frac{f(x)-f(\tilde x)}{x-\tilde x}\right|$ has an upper bound (because the absolute value is bounded, both positive and negative values are bounded). There exists a constant $$L\in\R$$, so that for all $$\tilde x, x \in D$$ with $$x\neq\tilde x$$ the inequality $\left|\frac{f(x)-f(\tilde x)}{x-\tilde x}\right|\leq L$ holds: This number $$L$$ is called the Lipschitz constant By multiplying both sides of the equation with $$|x-\tilde x|$$, we get:

"$\begin{align} && \left"

This inequality $$\left|f(x)-f(\tilde x)\right| \leq L |x-\tilde x|$$ is the basis for the definition of Lipschitz continuity. If we can find such a constant $$L$$ that satisfies this inequality for all $$x, \tilde x\in R$$, then we have also found an absolute bound for the rate of change of the function. Because the inequality $$\left|f(x)-f(\tilde x)\right| \leq L |x-\tilde x|$$ is still true even for $$x=\tilde x$$ this will result in a nice simplification of the definition,  because we can drop the assumption that $$x\neq\tilde x$$, which we otherwise would have needed, had we tried to work with the original mean rate of change $\frac{f(x)-f(\tilde x)}{x-\tilde x}$.

Definition
More formally the definition could be written using quantifiers:

"$\begin{array}{cc} & f:D\rightarrow\R \text{ is Lipschitz continuous } \\[1em] \iff& \exists L\geq 0\,\forall x,\tilde{x}\in D:"

The right-hand side of the above equivalence can be translated as follows:

"$ \underbrace{\exists L\geq 0}_{ \text{There exists } L\ge 0} \underbrace{\forall x,\tilde{x}\in D:}_{ \text{, so that for all } x,\tilde x \text{ following is true: }} \underbrace{|f(x)-f(\tilde{x})|\leq L\cdot |x-\tilde{x}|}_{ |f(x)-f(\tilde{x})| \text{ is smaller than } L\cdot |x-\tilde{x}|} $|undefined"

Why do we need Lipschitz continuity?
The Lipschitz constant of a Lipschitz continuous functions gives us an absolute upper bound for the rate of change. This is useful if we want to estimate function values.

Let us assume we are given a point $$x\in D$$ from the domain and the corresponding function value $$f(x)$$. Now imagine you want estimate the value $$f(y)$$ for a new point $$y\in D$$. We can obtain such an estimate if we use our Lipschitz constant to find an upper and lower bound for $$f(y)$$. From Lipschitz continuity it follows that:

"$\left"

By adding $$f(x)$$ we obtain:

"$f(x) - L"

This way we have found a lower and upper bound and know that $$f(y)$$ must be located in between.