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People often say that the derivative of a function at a point is its "best" linear (or affine) approximation around that point. This seems like a good intuition, but I've never seen it made precise. What I'd hope for is a sort of universal property, given a class of possible approximations, that specifies which approximation is the best one, if it exists.

I'd want to get continuity from the class of constant functions, derivatives from the class of affine functions, and in general the kth-order Taylor polynomial from the class of polynomials of degree up to k.

One possible definition of one approximation being better than another might be that its error is smaller at every point in some neighborhood. That is: Given some class $G$ of functions $: X \to Y$, $f : X \to Y$, and $x \in X$, The best $G$-approximation of $f$ at $x$ is a function $g_0 \in G$ such that for all $g \in G$, there's a neighborhood $A$ of $x$ such that for all $x' \in A$, $d(g_0(x'), f(x')) \le d(g(x'), f(x'))$.

(In particular, if we say that a function is continuous if it can be approximated by a constant function, we get this somewhat unusual definition of continuity: $f : X \to Y$ is continuous at $x \in X$ if for all $y \in Y$, there's a neighborhood $A$ of $x$ such that for all $x' \in A$, $d(f(x'), f(x)) \le d(f(x'), y)$.)

Does this definition do what I want? Can it be generalized to non-metric spaces? Or is there another definition of "best approximation" that works better?

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    $\begingroup$ A more tractable approach might be to say that its error is no greater (in the asymptotic sense) than any other approximation of the same type and then show that it is unique. $\endgroup$ – amd Apr 30 '18 at 23:19
  • $\begingroup$ To expand on @amd comment you can define define an order $\ll$ on function with $f \ll g$ iff $f=o(g)$. Then $f$ is continuous at $a$ if $f(x)-f(a) \ll C$ for any constant $C \neq 0$, differentiable with derivative $f'(a)$ if: $$f(x)-(f(a)+f'(a) (x-a)) \ll ax+b$$ for any $(a,b) \neq (0,0)$. The problem is that is becomes false for second derivative, take for example $x^2 \sin(1/x)$ on $0$. $\endgroup$ – Delta-u Apr 30 '18 at 23:23
  • $\begingroup$ Best approximation in this case means that the difference between a function and its linearisation around a given point decays quadratically as you approach the point. For all other linear functions the error decays at most linearly. $\endgroup$ – Michael Bächtold May 1 '18 at 11:38
  • $\begingroup$ For non affine spaces (like metric spaces or topological spaces, manifolds etc.) you should beware that the derivative of a map does not "live" on the same space, so it wouldn't directly make sense to compare the original map with the derivative. (On the other hand, from a perspective of synthetic differential geometry, the derivative of a map at a point is just the restriction of the map to an infinitesimal neighbourhood of the point.) $\endgroup$ – Michael Bächtold May 1 '18 at 11:59

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