Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces.
A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \}$ we have an $i$-linear bounded map $T_i \colon X^i \to Y$ such that: \begin{equation} \forall x \in X \quad P(x) = \sum_{i=0}^n T_i( \underbrace{ x, \ \ldots, \ x}_{i \text{ times}}) \end{equation} will be called an $n$-th degree polynomial. Note that here we allow $T_0$ to take any constant value.
We will say that the function $f \colon U \to Y$, where $U \subseteq X$, is $n$-approximable at $x \in U$ if there exists an $n$-th degree polynomial $P$ such that: \begin{equation} \lVert f(y) - P(y) \rVert_Y \in \omicron( \rVert y-x \rVert_X^n) \end{equation}
Question:
Let us suppose that $f \colon U \to Y$, where $U \subseteq X$, is open, is $n$-approximable at every point of its domain. What can be said about its regularity?
For example, since $1$-approximability at some point is equivalent to being Fréchet differentiable at that point, for $n=1$ we would get that $f \in D^1(U;Y)$, i.e., $f$ is differentiable at every point of its domain.
I would also be interested whether there are some results in more special cases, for example, when $X$ is finite-dimensional, or even when $X = \mathbb{R}$.
Note:
I think I might have an example of function $f \colon \mathbb{R} \to \mathbb{R}$ which is $n$-approximable for any $n \in \mathbb{N}$ but it isn't in $C^1(\mathbb{R})$.
Let $h \in C^\infty( \mathbb{R})$ be a convex non-negative, non-zero function such that its Taylor expansion at $0$ consists of only $0$. Now, I think there is a function $f \in C^\infty( \mathbb{R} \setminus \{0 \})$ (so it is $n$-approximable on this set for any $n \in \mathbb{N}$) and such that for some sequences $a_n, b_n \to 0$ we have $f'( a_n) = 1$ and $f'(b_n) = -1$, which also satisfies $-h(x) \leq f(x) \leq h(x)$ for all $x \in \mathbb{R}$ (so it is $n$-approximable at $0$ for any $n \in \mathbb{N}$).
Question 2:
Given the above note, I would like to ask a somewhat related question. Is there some notion of continuity of polynomials $T_i$ (viewed as a function of a single variable), which would give us a similar notion of regularity as $C^n$ for some $n \in \mathbb{N}$?
