Can we use linear map to approximate lipschitz continuous function $f$ in a compact domain after some linear transform?

Question

Suppose $f : \mathbb{R}\to \mathbb{R}$ is lipschitz continuous function , $K$ is a compact domain, for any $\varepsilon>0$, can we find $d,a\neq 0,c,w,b \in \mathbb{R}$ such that $\|df(ax+c)-(wx+b)\|<\varepsilon$ for any $x \in K$?

The intuitive understanding is to enlarge the function locally. Will it look like a straight line?

Is it necessary to add a differentiability requirement to $f$? Or $f$ only needs to be differentiable at a certain point, and then we can enlarge the neighborhood of that point to $K$ to achieve a linear approximation?

Answer 1

You don't even need Lipschitz continuity.

Trivial oversight

With your question statement, you can just choose $d = w = b = 0$ and $a$ and $c$ arbitrary. This holds for any function. So presumably you want $d \neq 0$.

Similarly, if you set $a = 0$, your identity holds with $w = 0$ and $b = d f(0)$. So probably you also want $a \neq 0$.

Slightly less trivial oversight

If $f:\mathbb{R}\to\mathbb{R}$ is continuous at $0$, and $K$ is compact set, then continuity at zero implies there exists $\delta > 0$ such that $y\in (-\delta,\delta) \implies |f(y) - f(0)| < \epsilon$. Since $K$ is compact, it is a subset of $[-R,R]$ for some $R > 0$. Choose then $w = 0 = c$, $b = f(0)$, and $a = \delta / (2R)$, and $d = 1$, then what you want holds.

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Based on the above I am pretty sure you asked the wrong question.

Since I mentioned differentiability, let me mention a couple things, which may be more along the line of what you are thinking about:

1. If $f$ is differentiable at a point $x_0$, then for every $\epsilon > 0$ and every compact set $K$ there exists an invertible affine transformation $A:\mathbb{R} \to \mathbb{R}$ and an affine function $B:\mathbb{R}\to\mathbb{R}$ such that $$ | f\circ A(x) - B(x) | \leq \epsilon |A(x)| $$ holds for all $x\in K$.
2. If $f$ is strongly differentiable at a point $x_0$, then for every $\epsilon > 0$ and every compact set $K$ there exists an invertible affine transformation $A:\mathbb{R}\to \mathbb{R}$ and a linear function $L:\mathbb{R}\to \mathbb{R}$ such that the Lipschitz seminorm $\| A^{-1} \circ f \circ A - L\| < \epsilon$ when the domain of the functions are restricted to $K$.