Functions near any Lipschitz

Question

Let $\operatorname{Lip}\subseteq C([0,1]^d,\mathbb{R}^d)$ be the set of all Lipschitz functions from $[0,1]^d$ to $\mathbb{R}^d$.

Which non-affine functions $f \in C(\mathbb{R}^d,\mathbb{R}^d)$ satisfy:

There exists some $\epsilon\geq \delta\triangleq K\epsilon>0$ such that, for every ${g} \in Lip$ and every $g_1,g_2 \in Ball_{\epsilon}({g})$ $$ \inf_{ \underset{B,A\in Mat_{d\times d}(\mathbb{R})}{a,b \in \mathbb{R}^d} } \sup_{x \in [0,1]^d} \big\|(a + A\, f (B\, g_1(x)+b)) - g_2(x)\big\|_2 < \delta. $$

Intuitively, I would expect that it's necessary for $f$ to be Lipschitz, maybe contractive.

Note: $Ball_{\epsilon}(g)\triangleq \left\{h \in C([0,1]^d;\mathbb{R}^d):\, \sup_{x \in [0,1]^d} \big\|h(x)- g(x)\big\|_2<\epsilon \right\}$ (for $\epsilon >0$ and $g \in C([0,1]^d;\mathbb{R}^d)$).

Answer 1

I don't have a complete characterization, but here's a pretty broad sufficient condition:

There are some nonsingular $T$ and $x_0 \in \mathbb{R}^d$ such that for any $\eta > 0$, there is some $r > 0$ such that $\|f(x) - f(x_0) - T(x - x_0)\| < r\eta$ in the ball $B_r(x_0)$.

Intuitively, this just says that $f$ can be rescaled to be arbitrarily close to affine. Also, it's worth noting that $T$ and $x_0$ are only fixed for convenience -- they need not be fixed so long as $\|T^{-1}\|$ is bounded.

The condition can be satisfied, for example, in the following "global" or "local" senses:

• There is some nonsingular affine map $S$ with $\|f(x) - S(x)\|$ bounded on $\mathbb{R}^d$.

• There is a point $x_0$ at which $f$ is differentiable with nonsingular total derivative (Jacobian).

These imply e.g. that it is not necessary for $f$ to be Lipschitz.

To prove our condition is sufficient, suppose we are given some $g, \varepsilon$. Fix $R > \sup_x \|g(x)\| + \varepsilon$, let $\eta > 0$, and take $r > 0$ satisfying our condition for $\eta$. Then setting $b = x_0$, $B = rR^{-1}I$, so $Bg_1(x) + b \in B_r(x_0)$ for $x \in [0, 1]^d$, and also setting $A = B^{-1}T^{-1}$, and $a = -B^{-1}b - Af(x_0)$, we have \begin{align*} \|a + Af(Bg_1(x) + b) - g_1(x)\| &= \|A(f(Bg_1(x) + b) - f(x_0) - T(Bg_1(x) + b))\| \\ &\leq \|A\| \|f(Bg_1(x) + b) - f(x_0) - T(Bg_1(x) + b)\| \\ &\leq \|A\| r \eta \\ &= R\|T^{-1}\|\eta \end{align*} so by the triangle inequality, $\inf_{a, b, A, B} \sup_x \|a + Af(Bg_1(x) + b) - g_2(x)\| \leq R\|T^{-1}\|\eta + 2\varepsilon$ for every $\eta > 0$, hence it's bounded by $\delta = 2\varepsilon$.