Can a data set of a function $f:\mathbb{R}^n\to\mathbb{R}^k$ decide if it is differentiable? Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}^k$ whose explicit formula is unknown, but there is a very large data set $\{x_i\}$ such that each $f(x_i)$ is known. Can we tell whether $f$ is differentiable at $x\in\mathbb{R}^n$?
I know there are approximation algorithms/theorems to $\textit{calculate}$ the derivative at each point, given the data set. However, I'm only interested in deciding whether they are at least $C^1$.
Context: if I have a collection $f_i:\mathbb{R}^n\to\mathbb{R}^k$ of such functions, I have their inverses (in a proper domain). I want to check whether the transition functions $f_i\circ f_j^{-1}$ are diffeomorphisms on the overlaps, defining a n-dimensional manifold embedded in $\mathbb{R}^k$. Common practice in mathematics is to do this with pencil and paper, but I wonder if we can automatize it.
 A: Since this question is tagged computational-science, I am assuming that the data set $\{x_i\}=\{x_1,\cdots,x_d\}$ is finite. If this is indeed the case, the answer is no. Even if $n=k=1$, we can always find, for example, a polynomial $p:\mathbb R\to \mathbb R$ so that $p(x_i)=f(x_i)$, and of course the polynomial is $C^1$. But We can also join each consecutive pair of points $(x_i,f(x_i))$, with a straight line to form a function that agrees with $p$ on every $x_i$, but which is not differentiable (unless all $(x_i,f(x_i))$ lie on a line). In fact, we could even build the function $$g(x)=p(x)+c(x)(x-x_1)\cdots(x-x_d),$$ where $c(x)$ is a nowhere-differentiable function. Then $g(x)$ is differentiable nowhere except possibly at each $x_i$.
From a purely mathematical standpoint, you cannot even infer that the function $f$ is continuous. Perhaps the function $f$ jumps to infinity quickly between $x_3$ and $x_4$ before becoming well-behaved again. Perhaps $f$ has a removable discontinuity at $x=1/\sqrt[5]{\pi}$. Perhaps $f$ is constant on each interval $[x_i,x_{i+1}),$ with jump discontinuities at the endpoints. All these examples are assuming $f:\mathbb R\to\mathbb R$, but the same types of behavior could also occur in higher dimensions.
