Continuous Function on Compact sets of the Real Line are Uniformly Continuous

Let $h,x,y,z$ be real values. Let $B(x,h) = (x-h,x+h)$ . One property of use here is $z \in B(y,g)$ and $y \in B(x,h)$ implies $z \in B(x,g+h)$ . Another property is $x \in B(y,h)$ iff $y \in B(x,h)$ .

Let $f : [0,1] \rightarrow \mathbb{R}$ be a continuous function. Then given $\epsilon > 0$ , for any $x \in [0,1]$ there is a $\delta_x > 0$ such that $f B(x,\delta_x) \subseteq B(fx,2^{-1}\epsilon)$ .

Let $\mathcal{B}$ be the collection of all $B(x,2^{-1} \delta_x)$ for $x \in [0,1]$ . Then $\mathcal{B}$ is an open cover of the closed and bounded set $[0,1]$ . Hence by the Heine-Borel theorem, there is a finite subcover of $\mathcal{B}$ . Let $x_1,\ldots,x_n$ denote the centres of the balls comprising this cover, and let $\delta = \min \{\delta_{x_i}:i=1..n \}$

Let $x,y\in [0,1]$ be such that $y\in B(x,2^{-1}\delta)$ . There is an $i \in \{1,\ldots,n\}$ such that $x\in B(x_i,2^{-1}\delta_{x_i})$ . Hence $y \in B(x_i,\delta_{x_i})$ . So $fx,fy \in B(fx_i,2^{-1}\epsilon)$ and so $fy \in B(fx,\epsilon)$ . Since $\delta$ is independent of $x$ , it follows that $f$ is uniformly continuous.

The uniform continuity property is a stronger property than continuity, since uniformly continuous functions are always continuous. When specialized to compact sets by the above result, the two concepts are equivalent.

The property of uniform continuity is important in investigations relating to commutativity of limiting operations with integration and differentiation.

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