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Let $f : X \to Y$ be a function between metric spaces with metrics $d_X, d_Y$. We might say that $f$ is "differentiable" at $x_0$ if: $$ \lim_{x \to x_0}\frac{d_Y(f(x), f(x_0))}{d_X(x,x_0)} $$ exists.

Is there anything interesting about such functions? Of course they don't satisfy differentiability in terms of "linear approximation" because there is no vector space structure on either space, but is there anything interesting/useful about such a definition?

rubikscube09
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  • Related: https://math.stackexchange.com/q/4573879/169085, https://math.stackexchange.com/q/4375527/169085 – Alp Uzman Oct 28 '23 at 21:57
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    Yes, you can even have e.g., Sobolev spaces (for $k=1$ arbitrary $p$, various versions) defined between $X,Y$. For example, see this book. Basically a 1-Lipschitz function can serve as a local coordinate so you can assume $Y$ is a Banach space, and you can use this to do various interesting things. – user10354138 Oct 28 '23 at 22:11
  • @user10354138 - very interesting, thank you for the link – rubikscube09 Oct 28 '23 at 22:26

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