Proof of The Gradient of a Twice Continuously Differentiable Function with Hessian Pinched between Two Positive Multiples of the Identity is a Bi-Lipschitz Bijection of Euclidean Space with Continuously Differentiable Inverse

Each result cited below is universally quantified over the data in its own statement.

Write $A(z)=D^{2}\Phi(z)$, the Hessian matrix, with entries $A_{jk}(z)=\partial_{j}\partial_{k}\Phi(z)$; it lies in $\mathcal{S}(d)$ and $\partial_{j}\partial_{k}\Phi=\partial_{k}\partial_{j}\Phi$ by claims 1 and 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian. Since $\Phi$ is of class $C^{2}$ on $\mathbb{R}^{d}$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives), clause 2 of C^k Maps on a Euclidean Open Set with $k=1$, read through the scalar convention of clause 3 there, shows that $\Phi$ and each $\partial_{k}\Phi$ are of class $C^{1}$ on $\mathbb{R}^{d}$; the $k$th component of $\nabla\Phi(x)=D\Phi(x)$ is $\partial_{k}\Phi(x)$ by Gradient of a Real-Valued Function on a Euclidean Open Set. By A Real-Valued C^1 Function is Differentiable at Every Point both $\Phi$ and each $\partial_{k}\Phi$ are differentiable at every point, the partial derivatives of $\partial_{k}\Phi$ being $\partial_{j}\partial_{k}\Phi$ (clause 4 of C^k Maps on a Euclidean Open Set). The $i$th coordinate of $z\in\mathbb{R}^{d}$ is $z\cdot e_{i}$ and $\lVert e_{i}\rVert=1$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis).

Claim 1 (Quadratic form and entry bounds). For all $z,v\in\mathbb{R}^{d}$ and $j,k\in[d]$,

By Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §ordering, $\varepsilon I_{d}\preceq A(z)\preceq LI_{d}$ means $v\cdot((\varepsilon I_{d})v)\le v\cdot(A(z)v)\le v\cdot((LI_{d})v)$, and $v\cdot((cI_{d})v)=c\,v\cdot v=c\lVert v\rVert^{2}$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). With $v=e_{j}$, Matrix-Vector Product gives $e_{j}\cdot(A e_{j})=A_{jj}$, so $\varepsilon\le A_{jj}\le L$ and $|A_{jj}|\le L$ (claim 6 of Properties of the Absolute Value in an Ordered Field, as $-L<0<\varepsilon$). For $j\ne k$ and $v=e_{j}\pm e_{k}$, bilinearity and $A_{jk}=A_{kj}$ give $v\cdot(Av)=A_{jj}+A_{kk}\pm2A_{jk}$, which is $\ge\varepsilon\lVert v\rVert^{2}\ge0$; hence $\mp A_{jk}\le\frac12(A_{jj}+A_{kk})\le L$, and $|A_{jk}|\le L$ by claim 6 of Properties of the Absolute Value in an Ordered Field.

Claim 2 (Monotonicity). For all $x,y\in\mathbb{R}^{d}$, with $h=y-x$,

Define $\gamma:[-1,2]\to\mathbb{R}$ by $\gamma(\tau)=\sum_{k=1}^{d}h_{k}\,\partial_{k}\Phi(x+\tau h)$. Let $\tau\in[0,1]$, an interior point of $[-1,2]$. By Chain Rule Along an Affine Path applied to $\partial_{k}\Phi$ (differentiable at $x+\tau h$), $\tau\mapsto\partial_{k}\Phi(x+\tau h)$ is differentiable at $\tau$ with derivative $\sum_{j}\partial_{j}\partial_{k}\Phi(x+\tau h)h_{j}$, so by Derivative of a Finite Linear Combination of Real Functions $\gamma$ is differentiable at $\tau$ with

by Matrix-Vector Product, the symmetry of $A$ and Claim 1. By Differentiability at an Interior Point Implies Continuity There, $\gamma$ is continuous at each $\tau\in[0,1]$ relative to $[-1,2]$, hence its restriction to $[0,1]$ is continuous on $[0,1]$; and at $\tau\in(0,1)$ the restriction is differentiable with the same derivative, directly from Derivative at an Interior Point (the condition there for the restriction concerns fewer increments). By Mean Value Theorem on a Closed Real Interval there is $\xi\in(0,1)$ with $\gamma(1)-\gamma(0)=\gamma'(\xi)\ge\varepsilon\lVert h\rVert^{2}$, and $\gamma(1)-\gamma(0)=(\nabla\Phi(y)-\nabla\Phi(x))\cdot h$.

Step 1 (Proof of lem:gradient-diffeomorphism-strongly-convex-2026a#bilipschitz). Let $x,y$, $h=y-x$. Lower bound: by Claim 2 and Cauchy-Schwarz Inequality for the Euclidean Dot Product (with claim 3 of Properties of the Absolute Value in an Ordered Field), $\varepsilon\lVert h\rVert^{2}\le\lVert\nabla\Phi(y)-\nabla\Phi(x)\rVert\,\lVert h\rVert$. If $h=0$ the lower bound is trivial; otherwise $0<\lVert h\rVert$ (claims 1 and 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and multiplying by $\lVert h\rVert^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\varepsilon\lVert h\rVert\le\lVert\nabla\Phi(y)-\nabla\Phi(x)\rVert$; as $\lVert x-y\rVert=\lVert h\rVert$ and $\lVert\nabla\Phi(x)-\nabla\Phi(y)\rVert=\lVert\nabla\Phi(y)-\nabla\Phi(x)\rVert$ (claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ with $\lambda=-1$), this is the left inequality. Upper bound: for each $k$, $\partial_{k}\Phi$ is of class $C^{1}$ on $\mathbb{R}^{d}$ with partial derivatives $\partial_{j}\partial_{k}\Phi=A_{jk}$ bounded in absolute value by $L$ (Claim 1), so part (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder, with $W=\mathbb{R}^{d}$ and $M_{1}=L$, gives $|\partial_{k}\Phi(y)-\partial_{k}\Phi(x)|\le\sqrt{d}\,L\lVert h\rVert$. Squaring (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both sides being nonnegative) and summing over $k$, claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ gives $\lVert\nabla\Phi(y)-\nabla\Phi(x)\rVert^{2}\le d\cdot d\,L^{2}\lVert h\rVert^{2}=(dL\lVert h\rVert)^{2}$, hence $\lVert\nabla\Phi(y)-\nabla\Phi(x)\rVert\le dL\lVert h\rVert$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field again.

Step 2 (Proof of lem:gradient-diffeomorphism-strongly-convex-2026a#bijection). Injectivity: if $\nabla\Phi(x)=\nabla\Phi(y)$, Step 1 gives $\varepsilon\lVert x-y\rVert\le0$, so $\lVert x-y\rVert=0$ and $x=y$ by claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$.

Surjectivity: fix $p\in\mathbb{R}^{d}$, let $\Psi(z)=\Phi(z)-p\cdot z$, and put $q=\nabla\Phi(0_{\mathbb{R}^{d}})-p$. The map $\Psi$ is continuous on $\mathbb{R}^{d}$ for $d_{E}$: $\Phi$ is, by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and $|p\cdot z-p\cdot z'|\le\lVert p\rVert\,\lVert z-z'\rVert$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product, so $z\mapsto p\cdot z$ is continuous, and claims 2 and 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space apply.

Fix $x\in\mathbb{R}^{d}$ and define $\phi(\tau)=\Phi(\tau x)-\tau\,(p\cdot x)=\Psi(\tau x)$ on $[-1,2]$. At each interior $\tau$, Chain Rule Along an Affine Path (base point $0_{\mathbb{R}^{d}}$, direction $x$) gives the derivative $\nabla\Phi(\tau x)\cdot x$ of $\tau\mapsto\Phi(\tau x)$, the map $\tau\mapsto\tau(p\cdot x)$ has derivative $p\cdot x$ (its difference quotients are constant, Derivative at an Interior Point), and Derivative of a Finite Linear Combination of Real Functions gives $\phi'(\tau)=(\nabla\Phi(\tau x)-p)\cdot x$. For $0<\tau$, Claim 2 with the points $0_{\mathbb{R}^{d}}$ and $\tau x$ gives $\tau(\nabla\Phi(\tau x)-\nabla\Phi(0_{\mathbb{R}^{d}}))\cdot x\ge\varepsilon\tau^{2}\lVert x\rVert^{2}$, and multiplying by $\tau^{-1}>0$ (claim 7 of Elementary Order Arithmetic in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field),

As in Claim 2, the restrictions of $\phi$ to $[0,\tfrac12]$ and $[\tfrac12,1]$ satisfy the hypotheses of Mean Value Theorem on a Closed Real Interval, giving $\xi_{1}\in(0,\frac12)$ and $\xi_{2}\in(\frac12,1)$ with $\phi(\frac12)-\phi(0)=\frac12\phi'(\xi_{1})\ge\frac12\,q\cdot x$ and $\phi(1)-\phi(\frac12)=\frac12\phi'(\xi_{2})\ge\frac12(q\cdot x+\frac{\varepsilon}{2}\lVert x\rVert^{2})$. Adding, and using $q\cdot x\ge-\lVert q\rVert\,\lVert x\rVert$ (Cauchy-Schwarz Inequality for the Euclidean Dot Product, claim 6 of Properties of the Absolute Value in an Ordered Field),

Let $R=4\varepsilon^{-1}\lVert q\rVert+1$, positive. If $\lVert x\rVert\ge R$, then $\frac{\varepsilon}{4}\lVert x\rVert-\lVert q\rVert\ge\frac{\varepsilon}{4}>0$ and $\lVert x\rVert\ge R>0$, so $\Psi(x)>\Psi(0_{\mathbb{R}^{d}})$ (claim 5 of Elementary Order Arithmetic in an Ordered Field).

Let $O=\{z:\lVert z\rVert<R\}$, bounded in $(\mathbb{R}^{d},d_{E})$ by Bounded Subset of a Metric Space (centre $0_{\mathbb{R}^{d}}$, radius $R$, using claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), and let $K$ be its closure, which is compact by The Closure of a Bounded Subset of $\mathbb{R}^n$ is Compact and contains $O\ni0_{\mathbb{R}^{d}}$ by claim 1 of The Closure is the Smallest Closed Superset. By Extreme Value Theorem on a Compact Subset of a Metric Space applied to the restriction of $\Psi$ to $K$ there is $x^{*}\in K$ with $\Psi(x^{*})\le\Psi(z)$ for all $z\in K$. Then $\Psi(x^{*})\le\Psi(z)$ for every $z\in\mathbb{R}^{d}$: if $\lVert z\rVert<R$ then $z\in K$; otherwise $\Psi(z)>\Psi(0_{\mathbb{R}^{d}})\ge\Psi(x^{*})$.

Fix $i\in[d]$ and let $g_{i}(s)=\Psi(x^{*}+se_{i})$ on the open interval $(-1,1)$. For $s\ne0$,

since $p\cdot e_{i}=p_{i}$; the point $x^{*}+se_{i}$ is $x^{*}$ with $i$th coordinate increased by $s$, so by Partial Derivative on a Euclidean Open Set and Derivative at an Interior Point, $g_{i}$ is differentiable at $0$ with $g_{i}'(0)=\partial_{i}\Phi(x^{*})-p_{i}$. As $g_{i}(0)\le g_{i}(s)$ for all $s$, $g_{i}$ has a local minimum at $0$ relative to $(-1,1)$, and Vanishing of the Derivative at an Interior Local Extremum gives $g_{i}'(0)=0$. Hence $\partial_{i}\Phi(x^{*})=p_{i}$ for every $i$, that is, $\nabla\Phi(x^{*})=p$.

Claim 3 (Continuity criterion). Let $g:\mathbb{R}^{d}\to\mathbb{R}$ and $a\in\mathbb{R}^{d}$, and suppose that for every real $\eta>0$ there is a real $\rho>0$ with $|g(y)-g(a)|<\eta$ for every $y$ with $\lVert y-a\rVert<\rho$. Then $g$ is continuous at $a$; the converse also holds. Indeed $\sum_{i=1}^{d}(y_{i}-a_{i})^{2}=\lVert y-a\rVert^{2}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $(g(y)-g(a))^{2}=|g(y)-g(a)|^{2}$ by claim 1 of Properties of the Absolute Value in an Ordered Field, and for nonnegative $s$ and positive $t$ we have $s<t$ if and only if $s^{2}<t^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; so the condition with $\delta=\rho$ is exactly that of Continuity at a Point for Maps Between Euclidean Spaces with $m=1$.

Claim 4 (Norm from coordinates). If $q\ge1$, $w\in\mathbb{R}^{q}$, $s$ is a nonnegative real and $|w_{k}|\le s$ for every $k\in[q]$, then $\lVert w\rVert\le q\,s$. Indeed $\lVert w\rVert=d_{E}(w,0_{\mathbb{R}^{q}})\le\sum_{k=1}^{q}|w_{k}|$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 1 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals, and $\sum_{k=1}^{q}|w_{k}|\le\sum_{k=1}^{q}s=q\,s$ by claims 2, 3 and 5 of Properties of Finite Sums applied to the nonnegative summands $s-|w_{k}|$.

Step 3 (Positive definiteness and the Jacobian of $\nabla\Phi$). Let $x\in\mathbb{R}^{d}$. The matrix $A(x)$ lies in $\mathcal{S}(d)$, and for $v\ne0_{\mathbb{R}^{d}}$ we have $0<\lVert v\rVert$ (claims 1 and 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), so $v\cdot(A(x)v)\ge\varepsilon\lVert v\rVert^{2}>0$ by Claim 1 and claim 5 of Elementary Order Arithmetic in an Ordered Field; thus $A(x)$ is positive definite. The components $\partial_{i}\Phi$ of $\nabla\Phi$ are of class $C^{1}$, and the entry of $D(\nabla\Phi)(x)$ in row $i$ and column $j$ is $\partial_{j}(\partial_{i}\Phi)(x)=\partial_{j}\partial_{i}\Phi(x)$ (clause 4 of C^k Maps on a Euclidean Open Set), which is $A_{ji}(x)=A_{ij}(x)$ by Hessian Matrix of a C^2 Function and claim 1 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian. Since real matrices with the same entries are equal (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices), $D(\nabla\Phi)(x)=D^{2}\Phi(x)$.

By Invertibility of Symmetric Positive Definite Matrices, $A(x)$ is invertible; write $B(x)=A(x)^{-1}$. For every $v\in\mathbb{R}^{d}$, claim 2 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product gives $A(x)(B(x)v)=(A(x)B(x))v=I_{d}v=v$ and $B(x)(A(x)v)=v$, where $(I_{d}v)_{i}=\sum_{j}\delta_{ij}v_{j}=v_{i}$ by Identity Matrix, Matrix-Vector Product and claim 7 of Properties of Finite Sums.

Step 4 (A bound for the inverse). For all $x,v\in\mathbb{R}^{d}$, $\lVert B(x)v\rVert\le\varepsilon^{-1}\lVert v\rVert$. Let $w=B(x)v$, so $A(x)w=v$ (Step 3). By Claim 1, claim 3 of Properties of the Absolute Value in an Ordered Field and Cauchy-Schwarz Inequality for the Euclidean Dot Product,

If $w=0_{\mathbb{R}^{d}}$ the bound is clear, as $\lVert w\rVert=0$ and $0\le\varepsilon^{-1}\lVert v\rVert$. Otherwise $0<\lVert w\rVert$, and multiplying by $\lVert w\rVert^{-1}$ and then by $\varepsilon^{-1}$, both positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, gives the bound by claim 5 of Elementary Arithmetic in an Ordered Field.

Step 5 (The inverse map is Lipschitz). By Step 2, $\nabla\Phi$ is a bijection of $\mathbb{R}^{d}$ onto $\mathbb{R}^{d}$; let $G=(\nabla\Phi)^{-1}$, so $\nabla\Phi(G(y))=y$ and $G(\nabla\Phi(x))=x$. For $y,y'\in\mathbb{R}^{d}$, Step 1 applied to $G(y)$ and $G(y')$ gives $\varepsilon\lVert G(y)-G(y')\rVert\le\lVert y-y'\rVert$, hence, multiplying by $\varepsilon^{-1}$ (claim 7 of Elementary Order Arithmetic in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field),

the second by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Given $y_{0}$ and $\eta>0$, every $y$ with $\lVert y-y_{0}\rVert<\varepsilon\eta$ satisfies $|G_{i}(y)-G_{i}(y_{0})|<\eta$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), so each $G_{i}$ is continuous at every point by Claim 3.

Step 6 (Differentiability of $\nabla\Phi$). Fix $x_{0}\in\mathbb{R}^{d}$ and write $A_{0}=A(x_{0})$. We show: for every real $\theta>0$ there is a real $\delta>0$ such that $\lVert\nabla\Phi(x_{0}+h)-\nabla\Phi(x_{0})-A_{0}h\rVert\le\theta\lVert h\rVert$ whenever $0<\lVert h\rVert<\delta$. For $k\in[d]$, since $\partial_{k}\Phi$ is of class $C^{1}$, A Real-Valued C^1 Function is Differentiable at Every Point shows that $\partial_{k}\Phi$ is differentiable at $x_{0}$ with derivative matrix the $1\times d$ matrix whose entry in column $i$ is $\partial_{i}\partial_{k}\Phi(x_{0})=(A_{0})_{ik}=(A_{0})_{ki}$ (Hessian Matrix of a C^2 Function, symmetry of $A_{0}$); the single coordinate of that matrix applied to $h$ is $\sum_{i}(A_{0})_{ki}h_{i}=(A_{0}h)_{k}$. On $\mathbb{R}^{1}$ the Euclidean norm is the absolute value: $\lVert u\rVert^{2}=u^{2}=|u|^{2}$ with both nonnegative (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 1 of Properties of the Absolute Value in an Ordered Field), so $\lVert u\rVert=|u|$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Hence Differentiability at a Point for Maps Between Euclidean Spaces, applied with the tolerance $\theta d^{-1}$, gives $\delta_{k}>0$ such that $0<\lVert h\rVert<\delta_{k}$ implies $|\partial_{k}\Phi(x_{0}+h)-\partial_{k}\Phi(x_{0})-(A_{0}h)_{k}|\le\theta d^{-1}\lVert h\rVert$. Let $\delta$ be the least of $\delta_{1},\dots,\delta_{d}$ (claim 9 of Elementary Order Arithmetic in an Ordered Field, applied repeatedly). For $0<\lVert h\rVert<\delta$, every coordinate of $\nabla\Phi(x_{0}+h)-\nabla\Phi(x_{0})-A_{0}h$ is bounded in absolute value by $\theta d^{-1}\lVert h\rVert$, so Claim 4 gives the asserted bound $d\cdot\theta d^{-1}\lVert h\rVert=\theta\lVert h\rVert$.

Step 7 (Differentiability of $G$). Fix $y_{0}\in\mathbb{R}^{d}$, let $x_{0}=G(y_{0})$, $A_{0}=A(x_{0})$ and $B_{0}=B(x_{0})$. Let $\eta>0$, put $\theta=\eta\varepsilon^{2}>0$, take $\delta$ from Step 6 for this $\theta$, and let $0<\lVert k\rVert<\varepsilon\delta$. Put $y=y_{0}+k$ and $h=G(y)-x_{0}$. As $y\ne y_{0}$ and $G$ is injective, $h\ne0_{\mathbb{R}^{d}}$, so $0<\lVert h\rVert$; by Step 5, $\lVert h\rVert\le\varepsilon^{-1}\lVert k\rVert<\delta$. Since $\nabla\Phi(x_{0}+h)=y$ and $\nabla\Phi(x_{0})=y_{0}$, Step 6 gives $\lVert r\rVert\le\theta\lVert h\rVert$ for $r=k-A_{0}h$. By claims 1 and 2 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product and Step 3, $B_{0}r=B_{0}k-B_{0}(A_{0}h)=B_{0}k-h$, so

By claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, Step 4, and claim 5 of Elementary Arithmetic in an Ordered Field,

As $\mathbb{R}^{d}$ is open (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and contains $y_{0}+k$, $G$ is differentiable at $y_{0}$ with derivative matrix $B_{0}$. By claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique, $\partial_{j}G_{i}(y_{0})$ exists and equals $(B_{0})_{ij}$ for all $i,j\in[d]$. Hence, for every $x\in\mathbb{R}^{d}$ (take $y_{0}=\nabla\Phi(x)$, so $x_{0}=x$), the matrix $DG(\nabla\Phi(x))$ has the same entries as $B(x)$, so it is the inverse matrix of $D^{2}\Phi(x)$.

Step 8 (Continuity of the partial derivatives of $G$). Fix $i,j\in[d]$ and $y_{0}\in\mathbb{R}^{d}$, with $x_{0}$, $A_{0}$, $B_{0}$ as in Step 7; by Step 7, $\partial_{j}G_{i}(y)=B(G(y))_{ij}$ for every $y$. For any real $d\times d$ matrix $M$, $(Me_{j})_{i}=\sum_{l}M_{il}(e_{j})_{l}=M_{ij}$ by Matrix-Vector Product, Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis and claim 7 of Properties of Finite Sums.

Let $\eta>0$ and $\theta=\eta\varepsilon^{2}(2d^{2})^{-1}>0$. For $k,l\in[d]$ the function $A_{kl}=\partial_{k}\partial_{l}\Phi$ is continuous at every point, by clause 1 of C^k Maps on a Euclidean Open Set applied to the $C^{1}$ function $\partial_{l}\Phi$, so by Claim 3 there is $\rho_{kl}>0$ with $|A_{kl}(x)-A_{kl}(x_{0})|<\theta$ whenever $\lVert x-x_{0}\rVert<\rho_{kl}$. Let $\rho$ be the least of the $\rho_{kl}$ (claim 9 of Elementary Order Arithmetic in an Ordered Field, applied repeatedly), and let $\lVert y-y_{0}\rVert<\varepsilon\rho$. With $x=G(y)$, Step 5 gives $\lVert x-x_{0}\rVert\le\varepsilon^{-1}\lVert y-y_{0}\rVert<\rho$. Write $A_{1}=A(x)$, $B_{1}=B(x)$ and $u=B_{0}e_{j}$; then $A_{0}u=e_{j}$ (Step 3) and $\lVert u\rVert\le\varepsilon^{-1}\lVert e_{j}\rVert=\varepsilon^{-1}$ (Step 4, Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis). By Step 3 and claims 1 and 2 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product,

the last step because $((A_{0}-A_{1})u)_{k}=\sum_{l}\bigl((A_{0})_{kl}-(A_{1})_{kl}\bigr)u_{l}=(A_{0}u)_{k}-(A_{1}u)_{k}$ by Difference of Real Matrices, Matrix-Vector Product and claims 2 and 3 of Properties of Finite Sums. By claim 3 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product, $\lVert(A_{0}-A_{1})u\rVert\le C\lVert u\rVert$ with $C=\lVert(c_{1},\dots,c_{d})\rVert$ and $c_{k}=\sum_{l}|(A_{0})_{kl}-(A_{1})_{kl}|$; here $0\le c_{k}\le d\theta$ by claims 2, 3 and 5 of Properties of Finite Sums, so $C\le d^{2}\theta$ by Claim 4. By Step 4 and claim 5 of Elementary Arithmetic in an Ordered Field,

Therefore, by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$,

and $\partial_{j}G_{i}$ is continuous at $y_{0}$ by Claim 3.

Step 9 (Proof of lem:gradient-diffeomorphism-strongly-convex-2026a#inverse). Step 3 gives the positive definiteness of $D^{2}\Phi(x)$ and $D(\nabla\Phi)(x)=D^{2}\Phi(x)$. For each $i\in[d]$, the component $G_{i}$ is continuous at every point (Step 5), and for each $j\in[d]$ its partial derivative $\partial_{j}G_{i}$ exists at every point (Step 7) and is continuous at every point (Step 8); by clause 1 of C^k Maps on a Euclidean Open Set, read through clause 3 there, $G_{i}$ is of class $C^{1}$ on $\mathbb{R}^{d}$. Finally, Step 7 shows that $DG(\nabla\Phi(x))$ is the inverse matrix of $D^{2}\Phi(x)$ for every $x\in\mathbb{R}^{d}$.