Proof of A $C^1$ Map into a Euclidean Space is Differentiable at Every Point

Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, used on $\mathbb{R}^{n}$, on $\mathbb{R}^{p}$ and on $\mathbb{R}^{1}$ alike, and $|t|$ for the absolute value of a real number $t$; the real numbers form an ordered field, whose order arithmetic is that of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field. Sums are finite sums in $\mathbb{R}$, and coordinates of differences of points are computed coordinatewise by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$.

Step 1 (each coordinate function is of class $C^{1}$). By clause 1 of C^k Maps on a Euclidean Open Set, for every $k$ with $1\le k\le p$ the coordinate function $f_k$ is continuous at every point of $U$, and for every $i$ with $1\le i\le n$ the partial derivative $\partial_i f_k$ exists at every point of $U$ and is continuous at every point of $U$. These are exactly the conditions of clause 1 of that definition for the map $f_k:U\to\mathbb{R}$, regarded through the scalar convention of clause 3 as a map into $\mathbb{R}^{1}$ with single coordinate function $f_k$; so each $f_k$ is of class $C^{1}$ on $U$. In particular all the partial derivatives $\partial_i f_k(a)$ exist, so by Jacobian Matrix of a Map Between Euclidean Spaces the Jacobian matrix $Df(a)$ is defined, with $p$ rows and $n$ columns and entry $\bigl(Df(a)\bigr)_{ki}=\partial_i f_k(a)$; likewise each $Df_k(a)$ is defined, with one row and $n$ columns and entry $\partial_i f_k(a)$ in column $i$.

We also record that for a point $z$ of $\mathbb{R}^{1}$ with single coordinate $t$ one has $\lVert z\rVert=|t|$: by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ the norm $\lVert z\rVert$ is the unique nonnegative real whose square is $t^{2}$, and $|t|$ is nonnegative with $|t|^{2}=t^{2}$ by claims 1 and 4 of Properties of the Absolute Value in an Ordered Field.

Step 2 (choice of constants). Let $N=\sum_{k=1}^{p}1$, a real number. Its summands are all equal to $1$, and $0\le 1$ by claim 1 of Elementary Arithmetic in an Ordered Field, so claim 6 of Properties of Finite Sums gives $1\le N$; since $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, claim 2 of that lemma gives $0<N$. By claim 7 of that lemma $N^{-1}$ exists and $0<N^{-1}$, and multiplying $1\le N$ by the nonnegative factor $N^{-1}$ using claim 5 of Elementary Arithmetic in an Ordered Field gives $N^{-1}\le N^{-1}N=1$.

Let $\varepsilon$ be a real number with $0<\varepsilon$, and put $\varepsilon'=\varepsilon\,N^{-1}$, which is positive by claim 5 of Elementary Order Arithmetic in an Ordered Field.

For each $k$ with $1\le k\le p$, A Real-Valued C^1 Function is Differentiable at Every Point says that $f_k$ is differentiable at $a$ with derivative matrix $Df_k(a)$, so by Differentiability at a Point for Maps Between Euclidean Spaces there is a real $\delta_k$ with $0<\delta_k$ such that every $h\in\mathbb{R}^{n}$ with $0<\lVert h\rVert<\delta_k$ satisfies $a+h\in U$ and

$$\bigl\lVert f_k(a+h)-f_k(a)-Df_k(a)\,h\bigr\rVert\le\varepsilon'\,\lVert h\rVert .$$

By repeated use of claim 9 of Elementary Order Arithmetic in an Ordered Field there is a real $\delta$ with $\delta\le\delta_k$ for every such $k$ and with $\delta$ equal to one of the $\delta_k$; in particular $0<\delta$.

Step 3 (the estimate). Fix $h\in\mathbb{R}^{n}$ with $0<\lVert h\rVert<\delta$. By claim 2 of Elementary Order Arithmetic in an Ordered Field we have $\lVert h\rVert<\delta_k$ for every $k$, so $a+h\in U$ and the displayed estimate of Step 2 holds for every $k$.

Put $z=f(a+h)-f(a)-Df(a)\,h$, a point of $\mathbb{R}^{p}$. By Matrix-Vector Product and the description of $Df(a)$ in Step 1, the $k$th coordinate of $Df(a)h$ is $\sum_{i=1}^{n}\partial_i f_k(a)\,h_i$, which is also the single coordinate of $Df_k(a)h$; and the $k$th coordinate of $f(a+h)-f(a)$ is $f_k(a+h)-f_k(a)$. Hence, using the identification of Step 1 between the norm on $\mathbb{R}^{1}$ and the absolute value,

$$|z_k|=\bigl\lVert f_k(a+h)-f_k(a)-Df_k(a)\,h\bigr\rVert\le\varepsilon'\,\lVert h\rVert\qquad\text{for every }k\text{ with }1\le k\le p .$$

Both $|z_k|$ and $\varepsilon'\lVert h\rVert$ are nonnegative, the latter by claim 5 of Elementary Arithmetic in an Ordered Field applied to $0\le\lVert h\rVert$, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|z_k|^{2}\le\bigl(\varepsilon'\lVert h\rVert\bigr)^{2}$, and $|z_k|^{2}=z_k^{2}$ by claim 4 of Properties of the Absolute Value in an Ordered Field. Summing over $k$ with claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, and using claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ on the left and claim 3 of Properties of Finite Sums on the right,

$$\lVert z\rVert^{2}=\sum_{k=1}^{p}z_k^{2}\le\sum_{k=1}^{p}\bigl(\varepsilon'\lVert h\rVert\bigr)^{2}=N\,\bigl(\varepsilon'\lVert h\rVert\bigr)^{2}.$$

Since $\varepsilon'=\varepsilon N^{-1}$, commutativity and associativity of multiplication give $N(\varepsilon'\lVert h\rVert)^{2}=N\,\varepsilon^{2}N^{-1}N^{-1}\lVert h\rVert^{2}=N^{-1}\bigl(\varepsilon^{2}\lVert h\rVert^{2}\bigr)$. Now $0\le\varepsilon^{2}\lVert h\rVert^{2}$, by claim 5 of Elementary Arithmetic in an Ordered Field applied twice starting from $0\le\lVert h\rVert^{2}$, so multiplying $N^{-1}\le 1$ by this nonnegative factor gives

$$N^{-1}\bigl(\varepsilon^{2}\lVert h\rVert^{2}\bigr)\le\varepsilon^{2}\lVert h\rVert^{2}=\bigl(\varepsilon\,\lVert h\rVert\bigr)^{2}.$$

Hence $\lVert z\rVert^{2}\le(\varepsilon\lVert h\rVert)^{2}$. Both $\lVert z\rVert$ and $\varepsilon\lVert h\rVert$ are nonnegative, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, read from right to left, gives

$$\bigl\lVert f(a+h)-f(a)-Df(a)\,h\bigr\rVert\le\varepsilon\,\lVert h\rVert .$$

Step 4 (conclusion). Thus for every real $\varepsilon$ with $0<\varepsilon$ there is a real $\delta$ with $0<\delta$ such that every $h\in\mathbb{R}^{n}$ with $0<\lVert h\rVert<\delta$ satisfies $a+h\in U$ and the last displayed inequality. Since $Df(a)$ is a real matrix with $p$ rows and $n$ columns, Differentiability at a Point for Maps Between Euclidean Spaces says exactly that $f$ is differentiable at $a$ with derivative matrix $Df(a)$.