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For a normed space $(V, \lVert\cdot\rVert_V)$ let us define: \begin{equation} \forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty. \end{equation}

I would like to ask whether the following characterization of (Fréchet) differentiability holds:

Let $(X, \lVert \cdot \rVert_X)$ be a normed space and $(Y, \lVert \cdot \rVert_Y)$ be a Banach space. Then, function $f \colon X \to Y$ is differentiable at $x \in X$ precisely when the following two conditions are satisfied:

  • $\limsup_{y \to x} \dfrac{ \lVert f(y) - f(x) \rVert_Y }{ \lVert y - x \rVert_X} \leq M$, for some $M < \infty$,
  • $ \forall y, z \in B(x, r) \quad \left\lVert f \circ \gamma_y^z - \gamma_{f(y)}^{f(z)} \right\rVert_{C([0,1]; Y)} \in o(r) $ as $r \to 0^+$.

The first condition is supposed to roughly correspond to the boundedness property of (Fréchet) differential, while the latter, to the linearity.

The reason as to why such a characterization might be interesting, is because it bypasses the need to define a differential. Hence, one could explain the regularity of a such a function without requiring to reference some other function between $X$ and $Y$.

Revision

After some thought, the initial second condition probably isn't enough to enforce differentiability. Hence, I'd like to re-ask the question with the second condition replaced with the following one. \begin{equation} \forall y,z \in X \; \; \forall t \in [0,1] \quad \lVert f \circ \gamma_y^z(t) - \gamma_{f(y)}^{f(z)}(t) \rVert_Y \in o \left( (1-t) \lVert y - x \rVert_X + t \lVert z - x \rVert_X \right). \end{equation}

The purpose of this change is that the previous condition only cared about \begin{equation} \max \left( \lVert y - x \rVert_X, \lVert z - x \rVert_X \right). \end{equation} Therefore, it wasn't able to detect that one of these distances might be much closer to $0$ than the other one (, in case of which, the secant connecting $(y, f(y))$ with $(z, f(z))$ should approximate graph much better around the argument closer to $x$) . The new one should be able to catch this property.

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    $\begingroup$ How would you define df (x) for a function satisfying these conditions then? $\endgroup$
    – Pietro Majer
    Commented Sep 19, 2021 at 4:19
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    $\begingroup$ @PietroMajer I think the differential $d_x f$ might be reconstructed from the "parametrised segments" $\gamma_{f(x)}^{f(y)}$ by some limiting process. The idea being that rays from $f(x)$ passing through $f(y)$ for $y$ in some neighbourhood of $x$ approach the graph of the differential (translated so that point $(0,0) \in X \times Y$ goes to $( x, f(x) )$ ) as the neighbourhood get smaller $\endgroup$
    – Kacper Kurowski
    Commented Sep 19, 2021 at 10:00
  • $\begingroup$ @PietroMajer Also, the idea of said limiting process is why I expect the assumption for $Y$ to be a Banach space to be useful. I think one would need to work with some Cauchy sequences and “build” the differential using their convergence. Moreover, I think that there might be a function, which has values in some normed space $Z$, which is not differentiable in $Z$, but differentiable in the completion of $Z$. Hence, making the assumption of the codomain being a Banach space necessary. $\endgroup$
    – Kacper Kurowski
    Commented Sep 19, 2021 at 10:13

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