Proof of Derivatives Along a Segment for $C^1$ Functions on a Euclidean Open Set

Throughout, $|\cdot|$ is the absolute value on $\mathbb{R}$, so that $d_{\mathbb{R}}(s,t)=|s-t|$ for real $s,t$ by The Absolute Value Metric on the Real Line. For $y,z\in\mathbb{R}^n$ write $Q(y,z)=\sum_{k=1}^n(y_k-z_k)^2$, the quantity appearing in Open Subset of Euclidean Space and in Continuity at a Point for Maps Between Euclidean Spaces, and set $s_h=\sum_{k=1}^n|h_k|$. Two elementary facts are used repeatedly. First, for $\tau,\sigma\in\mathbb{R}$,

Second, $\sum_{k=1}^nh_k^2\le s_h^2$, since expanding $s_h^2=\sum_{k}\sum_{l}|h_k||h_l|$ yields the terms $|h_k|^2=h_k^2$ together with nonnegative cross terms. In particular, if a real $\lambda>0$ satisfies $|\tau-\sigma|(1+s_h)<\lambda$, then $Q(x+\tau h,x+\sigma h)\le(\tau-\sigma)^2s_h^2<\lambda^2$, the last step by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Continuity of $f$ and of each $\partial_kf$ at every point of $W$, in the sense of Continuity at a Point for Maps Between Euclidean Spaces with $m=1$, is part of the hypothesis that $f$ is of class $C^1$ in C^k Maps on a Euclidean Open Set. For $m=1$ and a function $g$ that definition provides, for every real $\varepsilon>0$, a $\delta>0$ such that every $y$ in the domain with $Q(y,z)<\delta^2$ has $(g(y)-g(z))^2<\varepsilon^2$, and hence $|g(y)-g(z)|<\varepsilon$: indeed $|g(y)-g(z)|^2=(g(y)-g(z))^2<\varepsilon^2$, so $|g(y)-g(z)|\ge\varepsilon$ would contradict claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Derivative and partial-derivative values referred to below are single real numbers, by Uniqueness of the Derivative at an Interior Point and Uniqueness of the Partial Derivative on a Euclidean Open Set respectively.

Claim 1. If $h=0$, then $x+\tau h=x\in W$ for every $\tau$, and $r=1$ works. So assume $h\ne0$; then $s_h>0$.

Let $I=\{\tau\in\mathbb{R}:x+\tau h\in W\}$; the hypothesis says that every $\tau$ with $0\le\tau\le1$ lies in $I$. Every $\tau_0\in I$ has a surrounding interval inside $I$: since $W$ is open there is $\rho>0$ with $\{y:Q(y,x+\tau_0h)<\rho^2\}\subseteq W$, and then by the preliminary estimate every $\tau$ with $|\tau-\tau_0|<\rho/(1+s_h)$ lies in $I$; write $\eta(\tau_0)=\rho/(1+s_h)>0$ for one such choice.

Let $A=\{\tau\in\mathbb{R}:\tau>1,\ \tau\notin I\}$ and $B=\{\tau\in\mathbb{R}:\tau<0,\ \tau\notin I\}$.

If $A\ne\emptyset$: $A$ is bounded below by $1$, so $\beta=\inf A$ exists by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below. Every element of $A$ is $\ge1+\eta(1)$, since $\tau\in A$ with $\tau<1+\eta(1)$ would satisfy $|\tau-1|<\eta(1)$ and hence $\tau\in I$. Thus $1+\eta(1)$ is a lower bound of $A$, so $\beta\ge1+\eta(1)>1$. Moreover every $\tau$ with $1<\tau<\beta$ lies in $I$: otherwise $\tau\in A$, so $\beta\le\tau$, a contradiction. If $A=\emptyset$, set $\beta=2$; then again every $\tau$ with $1<\tau<\beta$ lies in $I$.

Symmetrically, if $B\ne\emptyset$, then $B$ is nonempty and bounded above by $0$, so it has a least upper bound $\gamma$ by the Dedekind completeness of $\mathbb{R}$ (The Real Numbers and Standard Notation). Every element of $B$ is $\le-\eta(0)$ (as above, using the interval around $0\in I$), so $\gamma\le-\eta(0)<0$, and every $\tau$ with $\gamma<\tau<0$ lies in $I$. If $B=\emptyset$, set $\gamma=-1$, with the same conclusion.

Let $r$ be the least of $1$, $\beta-1$, and $-\gamma$; it is positive as the least of three positive reals (claim 9 of Elementary Order Arithmetic in an Ordered Field, applied twice). Let $-r<\tau<1+r$. If $0\le\tau\le1$, then $\tau\in I$ by hypothesis; if $1<\tau<1+r$, then $\tau<\beta$ since $r\le\beta-1$, so $\tau\in I$; if $-r<\tau<0$, then $\gamma\le-r<\tau$, so $\tau\in I$. This proves claim 1.

Claim 2. Let $\tau_0$ be an interior point of $J$, and set $z=x+\tau_0h$ and $L=\sum_{i=1}^n\partial_if(z)\,h_i$.

If $h=0$, then $F$ has constant value $f(x)$ on $J$ and $L=0$, so $F$ is differentiable at $\tau_0$ with $F'(\tau_0)=0=L$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives. So assume $h\ne0$; then $s_h>0$.

Since $W$ is open and $z\in W$, fix $\rho>0$ such that $B_z=\{y\in\mathbb{R}^n:Q(y,z)<\rho^2\}\subseteq W$. Let $\varepsilon>0$ be real. For each $k\in\{1,\dots,n\}$, continuity of $\partial_kf$ at $z$ (preamble), applied with the positive real $\varepsilon/(n(1+|h_k|))$, gives $\delta_k>0$ such that every $y\in W$ with $Q(y,z)<\delta_k^2$ has

Let $\mu$ be the least of $\rho,\delta_1,\dots,\delta_n$ (positive, by repeated use of claim 9 of Elementary Order Arithmetic in an Ordered Field) and set $\delta=\mu/(1+s_h)>0$.

Let $t\in\mathbb{R}$ with $0<|t|<\delta$ and $\tau_0+t\in J$. For $k\in\{0,1,\dots,n\}$ let $w_k\in\mathbb{R}^n$ be the point whose $l$th coordinate is $z_l+th_l$ for $l\le k$ and $z_l$ for $l>k$; thus $w_0=z$ and $w_n=x+(\tau_0+t)h$. More generally, consider any point $y$ whose $l$th coordinate is $z_l+th_l$ for $l<k$, is $z_k+u$ with $|u|\le|t||h_k|$ for $l=k$, and is $z_l$ for $l>k$; then

so every such point, in particular each $w_k$, lies in $B_z\subseteq W$ and satisfies $Q(y,z)<\delta_{k'}^2$ for every $k'\in\{1,\dots,n\}$.

Now

Fix $k$ with $h_k\ne0$; if $h_k=0$ then $w_k=w_{k-1}$ and the $k$th term vanishes. Let $p$ be the lesser and $q$ the greater of $0$ and $th_k$, so $p<q$; let $K=\{u\in\mathbb{R}:p\le u\le q\}$, an interval; and define $g_k:K\to\mathbb{R}$ by $g_k(u)=f(y_k(u))$, where $y_k(u)$ is the point whose $l$th coordinate is $z_l+th_l$ for $l<k$, is $z_k+u$ for $l=k$, and is $z_l$ for $l>k$. By the preceding paragraph $y_k(u)\in B_z$ for every $u\in K$, so $g_k$ is well defined, with $g_k(0)=f(w_{k-1})$ and $g_k(th_k)=f(w_k)$.

First, $g_k$ is differentiable at every interior point $u$ of $K$ with $g_k'(u)=\partial_kf(y_k(u))$: the partial derivative of $f$ with respect to the $k$th variable exists at $y_k(u)\in W$ with value $\partial_kf(y_k(u))$ (the $C^1$ hypothesis), and for $w\in\mathbb{R}$ the point obtained from $y_k(u)$ by adding $w$ to its $k$th coordinate is exactly $y_k(u+w)$; hence the difference quotients of $g_k$ at $u$ coincide with those of Partial Derivative on a Euclidean Open Set, and the requirement of the derivative definition follows by restricting its quantifier to those $w$ with $u+w\in K$.

Second, $g_k$ is continuous at every point of $K$ relative to $K$: given real $\varepsilon'>0$, continuity of $f$ at $y_k(u)$ (preamble) gives $\delta'>0$ with $|f(y)-f(y_k(u))|<\varepsilon'$ whenever $y\in W$ and $Q(y,y_k(u))<\delta'^2$; for $v\in K$ with $d_{\mathbb{R}}(v,u)=|v-u|<\delta'$ one has $Q(y_k(v),y_k(u))=(v-u)^2<\delta'^2$, whence $d_{\mathbb{R}}(g_k(v),g_k(u))<\varepsilon'$.

By the mean value theorem applied to $g_k$ on $K$, there is $\xi_k$ with $p<\xi_k<q$ and $g_k(q)-g_k(p)=g_k'(\xi_k)\,(q-p)$. Whether $th_k$ is $q$ (for $th_k>0$) or $p$ (for $th_k<0$), this rearranges to

Summing over $k$ and dividing by $t\ne0$,

the terms with $h_k=0$ of both sums vanishing. Since $|\xi_k|\le|t||h_k|$, the point $y_k(\xi_k)$ satisfies $Q(y_k(\xi_k),z)<\mu^2\le\delta_k^2$, so

using $|h_k|<1+|h_k|$. As $\varepsilon>0$ was arbitrary and the estimate covers every $t$ with $0<|t|<\delta$ and $\tau_0+t\in J$, the function $F$ is differentiable at $\tau_0$ with $F'(\tau_0)=L$, which is claim 2.

Claim 3. Let $\sigma\in J$, write $z=x+\sigma h$, and let $g$ be either $f$ or $\partial_if$ for some $i\in\{1,\dots,n\}$, with $\Phi:J\to\mathbb{R}$ given by $\Phi(\tau)=g(x+\tau h)$. Given real $\varepsilon>0$, continuity of $g$ at $z$ (preamble) gives $\delta'>0$ such that every $y\in W$ with $Q(y,z)<\delta'^2$ has $|g(y)-g(z)|<\varepsilon$. Set $\eta=\delta'/(1+s_h)>0$. For $\tau\in J$ with $d_{\mathbb{R}}(\tau,\sigma)<\eta$, the preliminary estimate gives $Q(x+\tau h,z)<\delta'^2$, so $d_{\mathbb{R}}(\Phi(\tau),\Phi(\sigma))=|g(x+\tau h)-g(z)|<\varepsilon$. Hence $\Phi$ is continuous at $\sigma$ relative to $J$ in the sense of Continuous Map Between Metric Spaces, which gives claim 3 for $F$ and for each $\tau\mapsto\partial_if(x+\tau h)$.

Claim 4. Suppose $f$ is of class $C^2$ on $W$. By clause 2 of C^k Maps on a Euclidean Open Set (with $k=1$), each $\partial_if:W\to\mathbb{R}$ is of class $C^1$ on $W$, and by clause 4 there the partial derivative of $\partial_if$ with respect to the $j$th variable at any point of $W$ is $\partial_j\partial_if$ evaluated there. Fix an interior point $\tau_0$ of $J$. Applying claim 2 with $\partial_if$ in place of $f$ (the hypothesis that $x+\tau h\in W$ for every $\tau\in J$ is unchanged), the function $u_i:J\to\mathbb{R}$, $u_i(\tau)=\partial_if(x+\tau h)$, is differentiable at $\tau_0$ with

Since $G(\tau)=\sum_{i=1}^nh_i\,u_i(\tau)$ for $\tau\in J$, claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives shows that each constant multiple $h_iu_i$ is differentiable at $\tau_0$ with derivative $h_iu_i'(\tau_0)$, and then, by induction on the number of summands — the same claim applied to the sum of the first $i$ summands and the summand $h_{i+1}u_{i+1}$ — the function $G$ is differentiable at $\tau_0$ with

which is claim 4. $\blacksquare$