Is the following property suffictient for second order differentialbility?

Question

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ satisfying that $f(x+x_0)=T_{x_0}+o(\Vert x\Vert^2)$, then can we deduce that $f'$ is differentiable on $U$?

Answer 1

No. Let $$ f(x)=\begin{cases}x^3\sin\dfrac{1}{x} & \text{if }x\ne0,\\0 & \text{if } x=0.\end{cases} $$ Then $$ f'(x)=\begin{cases}3\,x^2\sin\dfrac{1}{x} -x\cos\dfrac1x& \text{if }x\ne0,\\0 & \text{if } x=0.\end{cases} $$ $f'$ is continuous and $f(x)=o(x^2)$, but $f'$ is not differentiable at $x=0$.