If I have a function $f:\mathbb{R}^3 \to \mathbb{R}$, can I write its value at the point $x \in \mathbb{R}^3$ in terms of a nearby point $y$ as follows: $$ f(x) = f(y) + \int_0^1 (x-y) \cdot \nabla f(y + t(x-y))dt $$ If so, what identity is this? Some kind of higher dimension fundamental theorem of calculus? If so, does $x$ even need to be near $y$?
Or is the above identity just completely invalid?
This is am immediate consequence of the fundamental theorem of calculus as we can define the function \begin{align}F_{x,y}(t):= f(y+t(x-y)) \end{align} for any $x,y\in \mathbb{R}^3$ and $t\in \mathbb{R}$ and obtain \begin{align}f(x)-f(y)=F_{x,y}(1)-F_{x,y}(0)=\int_0^1\partial_tF_{x,y}(t)\mathrm{d}t=\int_0^1\mathrm{d}f(y+t(x-y))(x-y)\mathrm{d}t \end{align} So it indeed holds for all $x,y\in \mathbb{R}^3$. It is also known as Taylor's theorem with integral remainder.