Proof of Coordinatewise Characterization of Continuity for Euclidean Maps

We first prove $(1)\Rightarrow(2)$. Assume that $f$ is continuous at $a$. Fix an index $j\in\{1,\dots,m\}$ and let $\varepsilon>0$. By continuity of $f$ at $a$, there exists $\delta>0$ such that whenever $x\in E$ and

one has

Since the $j$th summand is nonnegative, it follows that

Hence $|f_j(x)-f_j(a)|<\varepsilon$. This is exactly continuity of $f_j$ at $a$ in the sense of Continuity at a Point. Since $j$ was arbitrary, every coordinate function is continuous at $a$.

Now prove $(2)\Rightarrow(1)$. Assume that each coordinate function $f_j$ is continuous at $a$ in the sense of Continuity at a Point. Let $\varepsilon>0$. For each $j\in\{1,\dots,m\}$, continuity of $f_j$ at $a$ with tolerance $\varepsilon/m$ yields a number $\delta_j>0$ such that whenever $x\in E$ and

one has

Set

If $x\in E$ and

then in particular

so

Therefore,

because $m\in\mathbb{N}$ implies $m\ge 1$. Thus $f$ is continuous at $a$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces.