Confusion with definition of n times differentiability at a point

Question

Let $f: E \to \mathbb{R}$, $E\subset \mathbb{R}$ be a function. Then $f$ is differentiable at a point $x_0 \in E$ iff $\lim_{E \ni x \to x_0} \frac{f(x) - f(x_0)}{x-x_0}$ exists. We can consider function $f':E' \to \mathbb{R}$ where $E'$ consists of points where $f$ is differentiable. Then $f'$ also could be differentiable at $x_0$. For this we only need the existence of $\lim_{E' \ni x \to x_0} \frac{f'(x) - f'(x_0)}{x-x_0}$. For this we do not need $E'$ to contain some whole neighbourhood $U_E(x_0)$ of $x_0$ in set $E$, right? But for some reason I see in a lot of sources the contrary statement. For example according to those sources for $f$ to be $2$ times differentiable at $x_0$ it should be differentiable in some whole neighbourhood $U_E(x_0)$. I do not understand this requirement. For me it looks like additional requirement out of air. Could you please give me some good reason for this? Or do I miss something and my question does not make sense in general?

Answer 1

It depends on the definition of a differentiable function $f :E \to \mathbb R$. There are various approaches differing in assumptions on the domain $E$ of $f$ (see my answer to Is open set and interiority a necessary condition for differentiability?).

Most authors require the domain to be an open subset of $\mathbb R$, and then it is necessary to assume that $f'$ exists in an open neigborhood of $x_0$ if you want to have a chance that $f'$ is differentiable at $x_0$.

The most general approach is Tao's. He allows arbitrary $E \subset \mathbb R$ and defines differentiability of $f$ in each limit point $x_0$ of $E$.

Using this approach, we can consider the set $L^0(E)$ of limit points of $E$ and get a subset $D^1(E) \subset L^0(E)$ on which $f'(x)$ exists. Next we can consider the set $L^1(E) \subset D^1(E)$ of limit points of $D^1(E)$ and get a subset $D^2(E) \subset L^1(E)$ on which $f''(x)$ exists. We can iterate this process - but imo the resulting chain of subsets of $E$ is fairly unpleasant.

In order that $f^{(n)}(x_0)$ exists, we must require that $x_0$ is contained in all these sets, and this does not seem to be very transparent. It is much easier to require that there exists an open subset $U \subset \mathbb R$ with $x_0 \in U \subset E$ such that $f$ is $(n-1)$-times differentiable on $U$ and $f^{(n)}(x_0)$ exists.

Update.

With Tao's approach we can do it a bit more general by requiring the existence of an open subset $U \subset \mathbb R$ such that

1. $x_0 \in U \cap E$

2. All points of $U \cap E$ are limit points of $E$ (or equivalently, limit points of $U \cap E$)

3. $f$ is is $(n-1)$-times differentiable on $U \cap E$ and $f^{(n)}(x_0)$ exists.

This covers for example functions defined on closed or half-open intervals. The traditional approach is to work with left / right derivatives in the boundary points. With Tao's definition there is no need to use separate concepts for left and right derivatives.

Answer 2

We say that a function $f\colon \Omega \rightarrow \mathbb{R}$ defined on an arbitrary subset $\Omega \subset \mathbb{R}$ has a Whitney jet of order $n$ at a point $x_0 \in \Omega$, where $x_0$ is an accumulation point of $\Omega$, if there exist constants $A_1, \dots, A_n \in \mathbb{R}$ such that $$ f(x) = f(x_0) + A_1(x - x_0) + A_2(x - x_0)^2 + \dots + A_n(x - x_0)^n + r_n(x), $$ where $$ \lim_{\substack{x \to x_0 \\ x \in \Omega}} \frac{r_n(x)}{(x - x_0)^n} = 0. \tag{*} $$

We require that $x_0 \in \Omega$ be an accumulation point because the limit in $(*)$ must be taken over points of $\Omega$.

Note that if $\Omega$ is open, and $f$ is $n$-times differentiable in the classical sense, we have $$ A_k = \frac{f^{(k)}(x_0)}{k!}. \tag{**} $$

For an arbitrary set $\Omega$, for every accumulation point $x_0 \in \Omega$, the coefficients $A_k$ (if they exist) are uniquely determined, so one can define $f^{(k)}(x_0)$ via the identity $(**)$.

The polynomial $$ f(x_0) + A_1(x - x_0) + A_2(x - x_0)^2 + \dots + A_n(x - x_0)^n $$ is called a Whitney jet of $f$ of order $n$ at the point $x_0$.

To be fair, I did not found literature focused specifically on $n$-differentiable functions defined on arbitrary subsets of $\mathbb{R}$. However, there is an extensive and influential body of work concerning $C^n$-differentiable functions defined on closed subsets.

A particularly interesting question is the following: suppose $\Omega \subset \mathbb{R}$ is closed (for instance, a Cantor set), and suppose $f$ have Whitney jets of order $n$ at every point of $\Omega$, with all derivatives $f^{(k)}$ (for $k \leq n$) continuous on $\Omega$. Can $f$ be extended to a $C^n$ function on the whole real line $\mathbb{R}$, that is, a function that is $n$-times differentiable and such that $f^{(k)}$ is continuous for every $k\leq n$?

Such questions are quite subtle. They were studied in depth by Hassler Whitney, both for subsets of $\mathbb{R}$ and more generally for subsets of $\mathbb{R}^j$. The answer is positive if we assume additional compatibility conditions on the Whitney jets. See the references below for more details.

References

Hassler Whitney. Differentiable Functions Defined in Arbitrary Subsets of Euclidean Space Transactions of the American Mathematical Society, Vol. 40, No. 2 (Sep., 1936), pp. 309-317. https://doi.org/10.2307/1989869

Hassler Whitney. Differentiable Functions Defined in Closed Sets. I Transactions of the American Mathematical Society, Vol. 36, No. 2, 1934, 369-387. https://doi.org/10.2307/1989844

Hassler Whitney. Analytic Extensions of Differentiable Functions Defined in Closed Sets Transactions of the American Mathematical Society, Vol. 36, No. 1, 1934, pp. 63-89 (27 pages) https://doi.org/10.2307/1989708

Fefferman, Charles. A sharp form of Whitney’s extension theorem. Annals of Mathematics, 161(1), 509–577, 2005. https://doi.org/10.4007/annals.2005.161.509