Many continuous mappings are built coordinate by coordinate. The focus keyword coordinate mappings describes the method: prove continuity of scalar coordinate functions and then assemble the vector-valued map. This lesson also includes standard operators such as integration on $C[0,1]$ and mappings on discrete metric spaces. These examples prepare students for functional analysis, where operators between function spaces are studied through metric estimates. :::example[Coordinate Pair Mapping] Let $(X,d)$ be a metric space, and let $f_1:X\to\mathbb R$ and $f_2:X\to\mathbb R$ be continuous. Define $f:X\to\mathbb R^2$ by $f(x)=(f_1(x),f_2(x))$. Then $f$ is continuous. ::: :::proof Given that $f_1$ and $f_2$ are continuous. To prove that $f$ is continuous. Let $a\in X$ and $\epsilon>0$. Choose $\delta_1,\delta_2>0$ such that $|f_1(x)-f_1(a)|<\epsilon/\sqrt2$ when $d(x,a)<\delta_1$ and $|f_2(x)-f_2(a)|<\epsilon/\sqrt2$ when $d(x,a)<\delta_2$. If $\delta=\min\{\delta_1,\delta_2\}$ and $d(x,a)<\delta$, then $$\begin{equation*} \|f(x)-f(a)\|^2<\frac{\epsilon^2}{2}+\frac{\epsilon^2}{2}=\epsilon^2. \end{equation*}$$ Hence, $f$ is continuous. $\square$ ::: :::theorem[Coordinate Criterion in Euclidean Space] Let $f:X\to\mathbb R^n$ be given by $f(x)=(f_1(x),\ldots,f_n(x))$. Then $f$ is continuous if and only if every coordinate function $f_k$ is continuous. ::: :::proof Given the coordinate representation of $f$. To prove the equivalence. If $f$ is continuous and $x_m\to x$, then $\|f(x_m)-f(x)\|\to0$. Since $|f_k(x_m)-f_k(x)|\leq\|f(x_m)-f(x)\|$, each coordinate is continuous. Conversely, if all coordinates are continuous and $x_m\to x$, then $f_k(x_m)\to f_k(x)$ for every $k$. Hence $\|f(x_m)-f(x)\|\to0$. Therefore $f$ is continuous. $\square$ ::: :::example[Trigonometric Map into the Plane] The function $f:\mathbb R\to\mathbb R^2$ defined by $f(x)=(\cos x,\sin x)$ is continuous because both coordinate functions are continuous. ::: :::example[Stereographic Coordinate Mapping] The mapping $z\mapsto(2\operatorname{Re}z/(1+|z|^2),2\operatorname{Im}z/(1+|z|^2),(|z|^2-1)/(|z|^2+1))$ from $\mathbb C$ to $\mathbb R^3$ is continuous. Each coordinate is a quotient of continuous functions and the denominator $1+|z|^2$ is never zero. ::: :::example[Integral Operator on Continuous Functions] Let $X=C[0,1]$ with the uniform metric $d(x,y)=\sup_{0\leq t\leq1}|x(t)-y(t)|$. Define $\omega:X\to X$ by $(\omega(x))(t)=\int_0^t x(s)\,ds$. Then $\omega$ is continuous. ::: :::proof Given the integral operator $\omega$. To prove continuity. For $x,y\in C[0,1]$, $$\begin{align*} |\omega(x)(t)-\omega(y)(t)|&=\left|\int_0^t(x(s)-y(s))\,ds\right|\\ &\leq\int_0^t|x(s)-y(s)|\,ds\\ &\leq d(x,y). \end{align*}$$ Taking supremum over $t\in[0,1]$ gives $d(\omega(x),\omega(y))\leq d(x,y)$. Hence, $\omega$ is continuous. $\square$ ::: :::example[Functions on a Discrete Metric Space] Let $(X,d)$ be a discrete metric space and let $Y$ be any metric space. Every function $f:X\to Y$ is continuous, because choosing $\delta=1/2$ forces $d(x,a)<\delta$ to imply $x=a$. ::: :::exercise 1. Prove $x\mapsto(\sin x,x^2)$ is continuous. 2. Prove $z\mapsto(\operatorname{Re}z,\operatorname{Im}z)$ is continuous. 3. Show $x\mapsto2x$ is continuous on $C[0,1]$ with the uniform metric. ::: :::answer 1. Both coordinates are continuous. 2. The coordinate functions on $\mathbb C$ are continuous. 3. Since $d(2x,2y)=2d(x,y)$, the operator is continuous. :::