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](https://math.stackexchange.com/questions/3916802/)) 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.



Reference request: "Tangent relation" in metric spaces
One reference:



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> Elisabeth Burroni and Jacques Penon, [A metric tangential calculus](http://tac.mta.ca/tac/volumes/23/10/23-10abs.html). *Theory and Applications of Categories* 23 (2010), 199–220.
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The first sentence of the paper says that a fuller account of their work can be found in a longer (99-page) arXiv paper:



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> Elisabeth Burroni and Jacques Penon, [Elements for a metric tangential calculus](https://arxiv.org/abs/0912.1012). ArXiv:0912.1012, 2009.
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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.


