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.