Let $f$ be a function from $\mathbb{R}^n$ to $\mathbb{R}$. Then, I saw in a few engineering/physics books that
If $f$ is differentiable, then $f(x) = f(0) + \nabla f(0) \cdot x + O(|x|^2)$
But, isn't this statement, strictly speaking, not correct?
Don't we need the second derivatives of $f$ to exist to show that the remainder term is $O(|x|^2)$? If not, am I correct that all we can say is that the remainder is "liitle oh" $o(|x|^1)$