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Definition 1 A function $f:(X,{d}_{x})\to (Y,{d}_{y})$ is continuous at a point ${x}_{0}\in X$ if: for any $\u03f5>0$ there exists a $\delta >0$ such that if ${d}_{x}({x}_{0},{x}_{1})<\delta $ , then ${d}_{y}(f\left({x}_{0}\right),f\left({x}_{1}\right))<\u03f5$ .
Definition 2 A function: $f:(X,{d}_{x})\to (Y,{d}_{y})$ is uniformly continuous if: for every $\u03f5>0$ there exists a $\delta >0$ such that for all ${x}_{0}\in X$ : if ${d}_{x}({x}_{0},{x}_{1})<\delta $ , then ${d}_{y}(f\left({x}_{0}\right),f\left({x}_{1}\right))<\u03f5$ .
The difference between these definitions is that for a function to be simply continuous at a point, one need only find a $\delta $ for the given input ${x}_{0}$ , while for a function to be uniformly continuous, one needs to find for a given $\u03f5$ a single value of $\delta $ that works in the definition for every point over which the function is defined.
Example 1 Consider the function $f:(C\left[T\right],{d}_{\infty})\to (\mathbb{R},{d}_{0})$ defined as $f\left(x\right)=x\left({t}_{0}\right)$ . In words, the input to $f$ is a continuous function over $T$ , and the output from $f$ is the value of the input function evaluated at $t={t}_{0}$ . We ask the question: Is $f$ continuous at some function input ${x}_{1}\left(t\right)$ ?
This shows the continuity of $f\left(x\right)$ at ${x}_{1}\left(t\right)$ . However, because the selection of $\delta $ did not depend on the value of ${x}_{1}$ , we have also shown that the function $f$ is uniformly continuous.
Definition 3 The sequence of functions $\left\{{f}_{n}\right\}$ , ${f}_{n}:(X,{d}_{x})\to (Y,{d}_{y})$ converges pointwise to $f:(X,{d}_{x})\to (Y,{d}_{y})$ if: for each $x\in X$ , the sequence of values $\left\{{f}_{n}\left(x\right)\right\}$ converges to $f\left(x\right)$ in $(Y,{d}_{y})$ .
Definition 4 The sequence of functions $\left\{{f}_{n}\right\}$ , ${f}_{n}:(X,{d}_{x})\to (Y,{d}_{y})$ converges uniformly to $f:(X,{d}_{x})\to (Y,{d}_{y})$ if: for each $\u03f5>0$ , there exists ${n}_{0}\in {\mathbb{Z}}^{+}$ such that if $n\ge {n}_{0}$ , then ${d}_{y}(f\left(x\right),{f}_{n}\left(x\right))<\u03f5$ for all $x\in X$ .
The difference between these definitions is that for uniform convergence, there must exist a single value of ${n}_{0}$ that works in the definition of continuity for all possible values of $x\in X$ .
Example 2 Consider the sequence of functions ${x}_{n}\left(t\right):([0,1],{d}_{0})\to ([0,1],{d}_{0})$ given by ${x}_{n}\left(t\right)=\frac{t}{n}$ . One naturally suspects that ${x}_{n}\left(t\right)$ may be converging to the zero-valued function. We can show this formally as follows:
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