Let $f: D \subseteq \mathbb{R}^n \to \mathbb{R}$ be a scalar function on a set $D$.
Then let $\nabla f = \left(\dfrac{\partial f}{\partial x_i}\right)_{i = 1,…,N}$ be the corresponding gradient.
My question is, how is the domain of the gradient $\nabla f$ related to the domain of the function $f$, and under what assumption are they the same.
My thoughts:
Suppose that $f$ is differentiable in each of its arguments, then $\nabla f$ has domain equal to the interior of $D$. If $D$ was open, then the domain of $\nabla f$ is equal to $D$.
Is this all we can say about how these domains are related?
I am curious under what condition would the domain $\nabla f$ be a (super/sub)set of $D$.
Best Answer
The domain of the gradient is the domain of the function minus all the points where at least one of the derivatives does not exist.