Uncertainty analysis: bounded-error data confine the parameter of a $C^1$ prediction along a declared unit direction in ${\mathbb{R}}^n$

The interval $I$ is one of the four open forms of [F1], hence an open subset of $\mathbb{R}$; fix a real $\epsilon >0$ with $N_{\epsilon }(0)\subseteq I$. By [F2] the punctured neighbourhood $N_{\epsilon }^{\ast }(0)$ is a nonempty subset of $N_{\epsilon }(0)\subseteq I$, so $I$ contains $0$ and at least one further point and is therefore nondegenerate; by [F3] every point of $I$ is a limit point of $I$, that is, by [F4], $N_{\eta }^{\ast }(\theta )\cap I\mathrm{\ne }\mathrm{\varnothing }$ for every $\theta \in I$ and every real $\eta >0$.

The function $F$ is totally differentiable at $p:=\gamma (\theta )={\theta }_0+\theta u$, and $D_{u}F(p)=⟨\mathrm{\nabla }F(p),u⟩$: as $F$ is of class $C^1$, each first partial derivative ${\mathrm{\partial }}_{j}F$ exists and is continuous on the open set $U$ by [F16], so [F14] gives total differentiability of $F$ at $p\in U$, and [F17] gives $D_{u}F(p)=⟨\mathrm{\nabla }F(p),u⟩$.

The path $\gamma :I\to U$ is continuous at every ${\theta }^{\mathrm{\prime }}\in I$: given a real $\epsilon >0$ take $\delta :=\epsilon$; for ${\theta }^{\mathrm{\prime }\mathrm{\prime }}\in I$ with $\mathrm{\mid }{\theta }^{\mathrm{\prime }\mathrm{\prime }}-{\theta }^{\mathrm{\prime }}\mathrm{\mid }<\delta$ one has $d_2(\gamma ({\theta }^{\mathrm{\prime }\mathrm{\prime }}),\gamma ({\theta }^{\mathrm{\prime }}))=\parallel \gamma ({\theta }^{\mathrm{\prime }\mathrm{\prime }})-\gamma ({\theta }^{\mathrm{\prime }}){\parallel }_2=\parallel ({\theta }^{\mathrm{\prime }\mathrm{\prime }}-{\theta }^{\mathrm{\prime }})u{\parallel }_2=\mathrm{\mid }{\theta }^{\mathrm{\prime }\mathrm{\prime }}-{\theta }^{\mathrm{\prime }}\mathrm{\mid }\parallel u{\parallel }_2=\mathrm{\mid }{\theta }^{\mathrm{\prime }\mathrm{\prime }}-{\theta }^{\mathrm{\prime }}\mathrm{\mid }<\epsilon$, by [F19], the homogeneity axiom (N2) of [F22] and $\parallel u{\parallel }_2=1$; this is continuity in the metric sense of [F21].

The configuration displacement has Euclidean length $\parallel \gamma (\theta )-{\theta }_0{\parallel }_2=\parallel \theta u{\parallel }_2=\mathrm{\mid }\theta \mathrm{\mid }\parallel u{\parallel }_2=\mathrm{\mid }\theta \mathrm{\mid }$, by the Euclidean norm of [F19], the homogeneity axiom (N2) of [F22] and $\parallel u{\parallel }_2=1$.

For every $\theta \in I$ the components of the path satisfy ${\gamma }_j(\theta +t)={\gamma }_j(\theta )+t{u}_j$ for every real $t$ with $\theta +t\in I$, since ${\gamma }_j(\theta +t)=({\theta }_0{)}_j+(\theta +t)u_j=({\theta }_0{)}_j+\theta u_j+t{u}_j={\gamma }_j(\theta )+t{u}_j$ by field arithmetic; on $I$ the functions $1_I:=p_0{\mathrm{\mid }}_I$, constantly $1$, and ${\mathrm{i}\mathrm{d}}_I:=p_1{\mathrm{\mid }}_I$, the identity, are differentiable at $\theta$ with derivatives $0$ and $1$, because the power functions $p_0$ and $p_1$ have those derivatives at every real by [F7] and $\theta$ is a limit point of $I$ by step 1.1, so [F6] restricts; since ${\gamma }_j=({\theta }_0{)}_j\cdot 1_I+u_j\cdot {\mathrm{i}\mathrm{d}}_I$ on $I$, the sum and scalar rules of [F8] give that ${\gamma }_j$ is differentiable at $\theta$ with ${\gamma }_j^{\mathrm{\prime }}(\theta )=({\theta }_0{)}_j\cdot 0+u_j\cdot 1=u_j$.

Each composite ${\mathrm{\partial }}_{j}F\circ \gamma :I\to \mathbb{R}$ is continuous: for open $V\subseteq \mathbb{R}$ the preimage $({\mathrm{\partial }}_{j}F\circ \gamma {)}^{-1}[V]$ is, by [F23], the preimage ${\gamma }^{-1}[{\mathrm{\partial }}_j{F}^{-1}[V]]$; the set ${\mathrm{\partial }}_j{F}^{-1}[V]$ is open in the domain $U$ of the continuous ${\mathrm{\partial }}_{j}F$ by the choice-free equivalence (a)$\iff$(b) of [F21], and its preimage under the continuous $\gamma$ is therefore open in $I$; the same equivalence then gives continuity of ${\mathrm{\partial }}_{j}F\circ \gamma$ on $I$.

For every $\theta \in I$ the partial derivative of ${\gamma }_j$ in the direction $e_0=1$ equals its ordinary derivative, so ${\mathrm{\partial }}_0{\gamma }_j(\theta )=u_j$: by [F9] the directional derivative $D_{e_0}{\gamma }_j(\theta )$ is the limit at $0$ of $t\mapsto ({\gamma }_j(\theta +t)-{\gamma }_j(\theta ))\mathrm{/}t$, while by [F5] the derivative ${\gamma }_j^{\mathrm{\prime }}(\theta )=u_j$ of step 2.1 is the limit at $\theta$ of $x\mapsto ({\gamma }_j(x)-{\gamma }_j(\theta ))\mathrm{/}(x-\theta )$; the substitution $x=\theta +t$ carries one punctured domain bijectively onto the other and the two $\epsilon$-$\delta$ conditions of [F10] are the same condition, so $D_{e_0}{\gamma }_j(\theta )={\gamma }_j^{\mathrm{\prime }}(\theta )=u_j$.

For every $\theta \in I$ the partial derivative of the path is the vector ${\mathrm{\partial }}_0\gamma (\theta )=u$: the quotient $q_{\theta }(t):=(\gamma (\theta +t)-\gamma (\theta ))\mathrm{/}t$ on its domain $\{t:t\mathrm{\ne }0,\theta +t\in I\}$ is identically $u$ by the component identity in step 2.1, so its limit exists and defines ${\mathrm{\partial }}_0\gamma (\theta )$ by [F9], the $j$-th component of $q_{\theta }$ is $t\mapsto ({\gamma }_j(\theta +t)-{\gamma }_j(\theta ))\mathrm{/}t$ with limit ${\mathrm{\partial }}_0{\gamma }_j(\theta )$, and step 3.1 gives ${\mathrm{\partial }}_0{\gamma }_j(\theta )=u_j$; by the componentwise limit criterion [F11] the $j$-th coordinate of ${\mathrm{\partial }}_0\gamma (\theta )$ equals $u_j$ for every $j<n$, and two vectors in ${\mathbb{R}}^n$ with equal coordinates are equal by [F19], so ${\mathrm{\partial }}_0\gamma (\theta )=u$.

The only partial derivative of $\gamma$ is the constant ${\mathrm{\partial }}_0\gamma \equiv u$ on $I$ by step 4.1 applied at every $\theta \in I$, and it is continuous at $\theta$: given a real $\epsilon >0$, every $\delta >0$ serves for the continuity condition of [F12], because $\parallel {\mathrm{\partial }}_0\gamma ({\theta }^{\mathrm{\prime }})-{\mathrm{\partial }}_0\gamma (\theta ){\parallel }_2=\parallel u-u{\parallel }_2=0<\epsilon$ for every ${\theta }^{\mathrm{\prime }}\in I$.

The path $\gamma$ is totally differentiable at $\theta$ with $D\gamma (\theta )(1)=u$: the partial derivative ${\mathrm{\partial }}_0\gamma$ exists on the neighbourhood $I$ of $\theta$ in the open domain $I\subseteq \mathbb{R}$ and is continuous at $\theta$ by step 5.1, so $D\gamma (\theta )$ exists by [F14]; moreover $D\gamma (\theta )(1)=D\gamma (\theta )(e_0)={\mathrm{\partial }}_0\gamma (\theta )=u$ by [F15] and step 4.1.

The composite $g:=F\circ \gamma :I\to \mathbb{R}$ is totally differentiable at every $\theta \in I$ with $Dg(\theta )=DF(\gamma (\theta ))\circ D\gamma (\theta )$ by the chain rule [F20], applied to the totally differentiable $\gamma$ of step 6.1 and $F$ of step 1.2; consequently $g^{\mathrm{\prime }}(\theta )=Dg(\theta )(1)=DF(\gamma (\theta ))(D\gamma (\theta )(1))=DF(\gamma (\theta ))(u)=D_{u}F(\gamma (\theta ))=⟨\mathrm{\nabla }F(\gamma (\theta )),u⟩\ge m$ for every $\theta \in I$, where the first equality identifies the ordinary derivative with the total derivative applied to $1$ (the limits of [F9] and [F5] coincide as in step 3.1) and the remaining equalities are step 6.1, [F15] and step 1.2.

The derivative $g^{\mathrm{\prime }}$ is continuous on $I$: on $I$ one has $g^{\mathrm{\prime }}(\theta )={\sum }_{j<n}{u}_j({\mathrm{\partial }}_{j}F\circ \gamma )(\theta )$ by step 7.1, the gradient definition [F18] and the inner product of [F19]; the composites are continuous by step 2.2, the dictionary of [F21] transfers this to the real reading, and finite sums and scalar multiples of functions continuous at each point of $I$ are continuous there by the algebra clause of [F21]; in particular $g^{\mathrm{\prime }}$ is continuous on $I$ in the sense required by [F24].

The value of $g$ at the centre is $g(0)=F({\theta }_0+0\cdot u)=F({\theta }_0)=0$, and $I$ has at least two elements with $0\in I$ by step 1.1; the function $g:I\to \mathbb{R}$ is differentiable at every point of $I$ with continuous derivative by steps 7.1 and 8.1, and $g^{\mathrm{\prime }}\ge m>0$ on $I$ by step 7.1.

The bounded-error model comparison [F24] now applies to $g$ on $I$ with centre ${\theta }_0:=0$ and the recorded scalar $\widehat{y}$, $s$, $\delta$ of the statement, because a model value $y=g(\theta )=F({\theta }_0+\theta u)$ at the candidate $\theta \in I$ is exactly the quantity the calibration bound constrains; its conclusion is $\mathrm{\mid }\theta -0\mathrm{\mid }\le (s+\delta )\mathrm{/}m$, that is $\mathrm{\mid }\theta \mathrm{\mid }\le (s+\delta )\mathrm{/}m$, and the equivalent contrapositive form says that a proposed $\theta$ violating this bound is incompatible with the two stated bounds; with the displacement identity of step 1.4 the statement is proved.