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.


2 Answers 2


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.

  • $\begingroup$ This looks perfect! I will go through the paper. Thank you, Tom. $\endgroup$ Nov 28, 2022 at 18:47

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.

  • $\begingroup$ I only found the article "The various definitions of the derivative in linear topological spaces" in Russian, so I cannot read it. :/ $\endgroup$ Nov 17, 2022 at 13:59
  • $\begingroup$ The relation doesn't change when we replace the metric by strongly equivalent ones, right? (en.wikipedia.org/wiki/Equivalence_of_metrics) $\endgroup$ Nov 17, 2022 at 14:14
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    $\begingroup$ @Martin Brandenburg: The Russian Mathematical Surveys (RMS) English translation of the paper is online here, but it's behind a paywall. Bound volumes of the journal itself can be found in the library of most every U.S. university that has a Ph.D. program in math (at least volumes from the 1960s to 1990s; after this, budget cuts resulting in library subscription cuts make "most every" less applicable), but I don't know how widely available RMS is in other countries. $\endgroup$ Nov 17, 2022 at 15:03
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    $\begingroup$ I am not sure that the objection not being topological is decisive. There are many properties which refer to uniformity or the Lipschitz structure of metric spaces. $\endgroup$ Nov 17, 2022 at 18:05
  • $\begingroup$ "The relation doesn't change ..." @Martin Brandenburg: Obviously it is so, but then even for maps between normed spaces you get different tangency classes, each corresponding to some equivalence class of mutually strongly equivalent metrics. $\endgroup$
    – TaQ
    Nov 17, 2022 at 18:57

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