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At the moment, in my very limited knowledge of mathematics (=my first year at college), I've just seen this concept applied to prove that a function is Riemman integrable. I'd like to know if this concept was motivated for some special reason, or what is the importance in further studies of mathematics.

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    $\begingroup$ Cauchy didn't know it. $\endgroup$ – Tony Piccolo Aug 1 '13 at 7:02
  • $\begingroup$ @TonyPiccolo was it uniform continuity or uniform convergence? Uniform continuity predates Cauchy for many years. $\endgroup$ – OR. Aug 1 '13 at 7:04
  • $\begingroup$ @RGB: I quote Flett Differential analysis pag. 151 "... Cauchy failed to perceive the difference between continuity and uniform continuity ..." $\endgroup$ – Tony Piccolo Aug 1 '13 at 7:20
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    $\begingroup$ By the way the Riemann integrability of a continuous function can be proved without using uniform continuity link. $\endgroup$ – Tony Piccolo Aug 2 '13 at 7:02
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    $\begingroup$ It seems that Bolzano had grasped the distinction between continuity and uniform continuity link, before the Weierstrassian era. $\endgroup$ – Tony Piccolo Aug 2 '13 at 7:21
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It is a stronger version of continuity. It is most useful on open intervals, of course (by the Heine-Cantor theorem, a continuous function over a closed bounded interval is uniformly continuous). Many theorems that relate continuous functions and limits/integrability/etc can be extended to uniformly continuous functions over open/unbounded intervals. Many theorem that should "obviously" be true about continuous functions are only true about uniformly continuous functions, for example: "if $\int_0^\infty f(x) \mathrm d x = 0$, then $f(x) \to_{x\to\infty} 0$ is only true for uniformly continuous functions.

Another often used property is the following: if $A \subset X$ is dense, $X$ is a complete metric space, and $f : A \to Y$ is uniformly continuous, then $f$ can be extended to all of $X$.

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I the analysis of functions there are properties that we look for that are local and some other global. For example, being continuous is local, it doesn't care if you modify the function outside some neighborhood of the point. Being convex, is convex is kind of global, it involves all points in the domain where convexity occurs. Often you want to pass information from local to global properties. For example, you may know that a function is bounded in a neighborhood of every point (a local information) and you may want to know if the function is bounded on all its domain. This is not true in general.

Since it is still important sometimes to detect boundedness from local boundedness, you can study what additional properties of the function, or its domain can give you a positive answer.

The general property that typically can be checked for the domain of the function is being compact. Compact is a property that help in passing information from local properties to global properties. A function that is locally bounded on a compact is bounded. A continuous function is locally bounded, so a continuous function on a compact is bounded.

But what about imposing a property on the function instead. Bounded is the global version of locally bounded. Uniformly continuous is the global version of continuous.

Like for bounded and locally bounded, if the domain is compact then continuous implies uniformly continuous.

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One motivation comes from non-standard analysis, i.e. analysis with hyperreal numbers. This view is actually very useful (makes things obvious) when looking at e.g. the uniform limit theorem (the relationship to uniform convergence).

Here, a real function $f$ is continuous at $x$ if $\hat{f}(x+\varepsilon)-\hat{f}(x)$ is infinitesimal for all infinitesimal $\varepsilon$.

A real function is uniformly continuous if it is continuous for all hyperreal $x$ -- whereas a continuous function only needs to be continuous at real values of $x$.

So it's obvious why $x^2$ is not uniformly continuous -- at $\omega$, it turns increments by $1/\omega$ into increments by $1$. Or why $1/x$ isn't uniformly continuous on the positive reals -- at $\varepsilon$, it turns increments by $\varepsilon$ into increments by $1/\varepsilon$. It also explains why $\sqrt{x}$ is continuous on the positive reals -- although it turns $\varepsilon$ into $\sqrt{\varepsilon}$, which has "higher order" -- that's still an infinitesimal.

In real number speak, this just says that for any two sequences st. $x_n-y_n\to 0$, $f(x_n)-f(y_n)\to 0$ (which is really the "sequential" form of stating uniform continuity). By contrast for continuity, this is only required with constant sequences $y_n$.

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