Reference request: "Tangent relation" in metric spaces

Question

Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there is some $\delta > 0$ such that for all $x \in X$ we have $$d(x,x_0) \leq \delta \implies d(f(x),g(x)) \leq \epsilon \cdot d(x,x_0).$$ In this case, let's write $f \sim_{x_0} g$. An equivalent characterization is $$ f(x_0)=g(x_0) \quad \text{and} \quad \lim_{x \to x_0,\, x \neq x_0} \frac{d(f(x),g(x))}{d(x,x_0)} = 0.$$ This relation appears (in the special case of normed vector spaces) in Dieudonné's "Foundations of Modern Analysis" to give a conceptual definition of the Fréchet derivative.

Question. What are good references for the properties of this tangent relation? Is it, perhaps, studied under a different name (touching, closeness, etc.)? In particular, I would like to know if the following results about the compatibility with composition are already in the literature?

(1) Let $X,Y,Z$ be metric spaces, $x_0 \in X$, $y_0 \in Y$. Let $g_1,g_2 : Y \to Z$ be functions with $g_1 \sim_{y_0} g_2$, and let $f : X \to Y$ be a function which is Lipschitz continuous at $x_0$ (def. here) with $f(x_0)=y_0$. Then also $g_1 \circ f \sim_{x_0} g_2 \circ f$.

(2) Let $X,Y,Z$ be metric spaces, $x_0 \in X$. Let $f_1, f_2 : X \to Y$ be functions with $f_1 \sim_{x_0} g_2$. If $g : Y \to Z$ is Lipschitz continuous, then also $g \circ f_1 \sim_{x_0} g \circ f_2$.

They are not hard to prove, but what makes them interesting is that they offer a very conceptual (new?) proof of the chain rule of the Fréchet derivative.

Answer 1

One reference:

Elisabeth Burroni and Jacques Penon, A metric tangential calculus. Theory and Applications of Categories 23 (2010), 199–220.

The first sentence of the paper says that a fuller account of their work can be found in a longer (99-page) arXiv paper:

Elisabeth Burroni and Jacques Penon, Elements for a metric tangential calculus. ArXiv:0912.1012, 2009.

Remark 1.6(2) of the first of these papers looks like it might cover the two specific properties you mention, but I haven't checked carefully.

Answer 2

I don't know whether there are "good" references to what you are actually asking but at least in some kind of implicit sense this kind of "tangency" was already considered by Fréchet in the first half of the 20th centure before it was noticed that it is not a good concept for the definition of derivative for maps between vector spaces since it is not topologically invariant but depends on the metrics used. For example, consider maps $E=\mathbb R\to\mathbb R=F$ with $F$ equipped with the standard absolute value metric and $E$ with the metric $(s,t)\mapsto\sqrt{|s-t|}$. Then the identity is tangent to the zero map at zero. This kind of note was already made in the article by V. I. Averbukh and O. G. Smolyanov: `The various definitions of the derivative in linear topological spaces´ in Russian Math. Surveys 23;4 (1968) pp. 67−113. See in particular the discussion there on page 76.