Proof of Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set

Throughout, $|\cdot|$ denotes the absolute value on $\mathbb{R}$, and $\mathbb{R}$ is regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line and $\mathbb{R}^n$ as a metric space through the Euclidean distance $d_E$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$. For $a=(a_1,\dots,a_n)\in U$, an index $i\in\{1,\dots,n\}$ and $s\in\mathbb{R}$, let $a[s]$ denote the point of $\mathbb{R}^n$ whose $i$th coordinate is $s$ and whose $k$th coordinate is $a_k$ for every $k\ne i$, so that $a[a_i]=a$; this is the notation of Slice Function and the Partial Derivative, and the difference quotients of the definition of the partial derivative are formed at the points $a[a_i+t]$. Given two functions $g,h:U\to\mathbb{R}$, we write $g-h$ for the function on $U$ whose value at $y$ is $g(y)-h(y)$.

Step 1 (partial derivatives of a difference). Let $g,h:U\to\mathbb{R}$, let $a\in U$ and $i\in\{1,\dots,n\}$, and suppose that the partial derivatives $\partial g/\partial x_i(a)$ and $\partial h/\partial x_i(a)$ exist, each definition being applied with $m=1$ and with the function as its single coordinate function. Then the partial derivative of $g-h$ with respect to the $i$th variable exists at $a$, and

Write $L=\partial g/\partial x_i(a)$ and $M=\partial h/\partial x_i(a)$. By claim 1 of Slice Function and the Partial Derivative there is $\rho\in\mathbb{R}$ with $0<\rho$ such that $a[s]\in U$ for every $s\in\mathbb{R}$ with $|s-a_i|<\rho$; in particular $a[a_i+t]\in U$ whenever $|t|<\rho$, so all the difference quotients below are defined.

Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. By claim 8 of Elementary Order Arithmetic in an Ordered Field we have $0<\varepsilon\cdot 2^{-1}$ and $\varepsilon\cdot 2^{-1}+\varepsilon\cdot 2^{-1}=\varepsilon$. By the definition of the partial derivative there are $\delta_1,\delta_2\in\mathbb{R}$ with $0<\delta_1$ and $0<\delta_2$ such that every $t\in\mathbb{R}$ with $0<|t|<\delta_1$ satisfies

and every $t\in\mathbb{R}$ with $0<|t|<\delta_2$ satisfies

By claim 9 of Elementary Order Arithmetic in an Ordered Field, applied twice, there is $\delta\in\mathbb{R}$ with $\delta\le\rho$, $\delta\le\delta_1$ and $\delta\le\delta_2$ and with $\delta$ equal to one of $\rho,\delta_1,\delta_2$; in particular $0<\delta$.

Let $t\in\mathbb{R}$ with $0<|t|<\delta$. By claim 2 of Elementary Order Arithmetic in an Ordered Field we get $|t|<\rho$, $|t|<\delta_1$ and $|t|<\delta_2$, so both displayed estimates apply and $a[a_i+t]\in U$. Since $t\ne 0$, the field arithmetic of $\mathbb{R}$ gives

By claims 2 and 5 of Properties of the Absolute Value in an Ordered Field the absolute value of a difference $A-B$ satisfies $|A-B|\le|A|+|B|$, so the absolute value of the left-hand side is at most the sum of the two quantities estimated above. By claim 3 of Elementary Order Arithmetic in an Ordered Field that sum is smaller than $\varepsilon\cdot 2^{-1}+\varepsilon\cdot 2^{-1}=\varepsilon$, and by claim 2 of that lemma we conclude

As $\varepsilon$ was arbitrary, the real number $L-M$ witnesses the definition of the partial derivative of $g-h$ with respect to the $i$th variable at $a$, which proves Step 1.

Step 2 (continuity of a difference). Let $g,h:U\to\mathbb{R}$ and $a\in U$, and suppose that $g$ and $h$ are continuous at $a$ in the Euclidean sense, each regarded as a map into $\mathbb{R}^m$ with $m=1$. Then $g-h$ is continuous at $a$ in the Euclidean sense.

By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, applied with $E=U$, the functions $g$ and $h$ are continuous at $a$ relative to $U$ as maps into $(\mathbb{R},d_{\mathbb{R}})$. Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. By claim 8 of Elementary Order Arithmetic in an Ordered Field, $0<\varepsilon\cdot 2^{-1}$ and $\varepsilon\cdot 2^{-1}+\varepsilon\cdot 2^{-1}=\varepsilon$, so there are $\delta_1,\delta_2\in\mathbb{R}$, both positive, such that every $y\in U$ with $d_E(a,y)<\delta_1$ satisfies $d_{\mathbb{R}}(g(y),g(a))<\varepsilon\cdot 2^{-1}$ and every $y\in U$ with $d_E(a,y)<\delta_2$ satisfies $d_{\mathbb{R}}(h(y),h(a))<\varepsilon\cdot 2^{-1}$. By claim 9 of Elementary Order Arithmetic in an Ordered Field let $\delta$ be the smaller of $\delta_1$ and $\delta_2$; then $0<\delta$. Let $y\in U$ with $d_E(a,y)<\delta$. By The Absolute Value Metric on the Real Line the metric $d_{\mathbb{R}}$ is given by $d_{\mathbb{R}}(s,t)=|s-t|$, and the field arithmetic of $\mathbb{R}$ gives

By claims 2 and 5 of Properties of the Absolute Value in an Ordered Field and claims 2 and 3 of Elementary Order Arithmetic in an Ordered Field,

Hence $g-h$ is continuous at $a$ relative to $U$ as a map into $(\mathbb{R},d_{\mathbb{R}})$, and claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions converts this back into continuity at $a$ in the Euclidean sense. This proves Step 2.

Step 3 (proof of claim 1). By claim 1 of the definition of class $C^2$, both $u$ and $\varphi$ are of class $C^1$ on $U$. By the definition of class $C^1$ this means that $u$ and $\varphi$ are continuous at every point of $U$ in the Euclidean sense, that all the partial derivatives $\partial u/\partial x_i$ and $\partial\varphi/\partial x_i$ exist at every point of $U$, and that the resulting functions from $U$ to $\mathbb{R}$ are continuous at every point of $U$ in the Euclidean sense.

By Step 2, $u-\varphi$ is continuous at every point of $U$ in the Euclidean sense. By Step 1, for every $x\in U$ and every $i\in\{1,\dots,n\}$ the partial derivative $\partial(u-\varphi)/\partial x_i(x)$ exists and equals $\partial u/\partial x_i(x)-\partial\varphi/\partial x_i(x)$; that is, as functions on $U$,

Applying Step 2 to the two continuous functions $\partial u/\partial x_i$ and $\partial\varphi/\partial x_i$ shows that $\partial(u-\varphi)/\partial x_i$ is continuous at every point of $U$ in the Euclidean sense. Hence $u-\varphi$ is of class $C^1$ on $U$.

By claim 2 of the definition of class $C^2$, for each $i$ the functions $\partial u/\partial x_i$ and $\partial\varphi/\partial x_i$ are of class $C^1$ on $U$. Repeating the previous paragraph with $u$ and $\varphi$ replaced by $\partial u/\partial x_i$ and $\partial\varphi/\partial x_i$, and using the displayed identity of functions, we conclude that $\partial(u-\varphi)/\partial x_i$ is of class $C^1$ on $U$ for every $i$. Together with the preceding paragraph this shows that $u-\varphi$ is of class $C^2$ on $U$.

Now fix $x\in U$. By the definition of the gradient, the $i$th coordinate of $D(u-\varphi)(x)$ is $\partial(u-\varphi)/\partial x_i(x)=\partial u/\partial x_i(x)-\partial\varphi/\partial x_i(x)$, which by the definition of the difference of points of $\mathbb{R}^n$ is exactly the $i$th coordinate of $Du(x)-D\varphi(x)$. As this holds for every $i$, the two points of $\mathbb{R}^n$ coincide.

By the definition of the Hessian matrix, the entry of $D^2(u-\varphi)(x)$ in row $i$ and column $j$ is $\partial^2(u-\varphi)/\partial x_i\,\partial x_j(x)$, which by the notation introduced in C^2 Real-Valued Map on an Open Subset of Euclidean Space is the partial derivative with respect to the $i$th variable, at $x$, of the function $\partial(u-\varphi)/\partial x_j$ on $U$. By the displayed identity of functions that function is $\partial u/\partial x_j-\partial\varphi/\partial x_j$, so Step 1 gives

which by the definition of the difference of real matrices is exactly the entry of $D^2u(x)-D^2\varphi(x)$ in row $i$ and column $j$. As this holds for all $i$ and $j$, the two matrices coincide. This proves claim 1.

Step 4 (proof of claim 2). Let $c\in\mathbb{R}$ and let $a\in U$.

First, $k_c$ is continuous at $a$ in the Euclidean sense: given $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, take $\delta=\varepsilon$; every $x\in U$ satisfies $(k_c(x)-k_c(a))^2=(c-c)^2=0$, and $0<\varepsilon^2$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, so the required implication holds.

Second, fix $i\in\{1,\dots,n\}$. By claim 1 of Slice Function and the Partial Derivative there is $\rho\in\mathbb{R}$ with $0<\rho$ such that $a[s]\in U$ whenever $|s-a_i|<\rho$. Given $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, take $\delta=\rho$: for every $t\in\mathbb{R}$ with $0<|t|<\rho$ we have $a[a_i+t]\in U$ and

using claim 1 of Properties of the Absolute Value in an Ordered Field for $|0|=0$. Hence $\partial k_c/\partial x_i(a)$ exists and equals $0$.

Since $a\in U$ was arbitrary, $k_c$ is continuous at every point of $U$ and all its first partial derivatives exist on $U$, the function $\partial k_c/\partial x_i$ being the constant function $k_0$ for every $i$. The two paragraphs above, applied to $k_0$, show that $k_0$ is continuous at every point of $U$ and that all its first partial derivatives exist and vanish on $U$. Hence $k_c$ is of class $C^1$ on $U$, and each $\partial k_c/\partial x_i=k_0$ is of class $C^1$ on $U$, so $k_c$ is of class $C^2$ on $U$.

Finally, for $x\in U$ the $i$th coordinate of $Dk_c(x)$ is $\partial k_c/\partial x_i(x)=0$ by Gradient of a Real-Valued Function on a Euclidean Open Set, so $Dk_c(x)$ is the origin of $\mathbb{R}^n$; and the entry of $D^2k_c(x)$ in row $i$ and column $j$ is the partial derivative with respect to the $i$th variable, at $x$, of the function $\partial k_c/\partial x_j=k_0$, which is $0$, so $D^2k_c(x)$ is the $n\times n$ real matrix all of whose entries are $0$. This proves claim 2.