I have the following question regarding the connection between vector fields and gradient fields.
Assume I have given a vector field $$\textbf{F} : \mathbb{R}^n \to \mathbb{R}^n.$$ I want to know when there exists a potential function $f:\mathbb{R}^n \to \mathbb{R}$ such that $$ \nabla f = \textbf{F}.$$ If $\textbf{F}$ is a smooth the following Theorem holds.
$\textbf{Theorem:}$
Let $\textbf{F} = (f_1, \dots, f_n) \in C^1(\mathbb{R}^n, \mathbb{R}^n)$. Then $\textbf{F}$ is a gradient field if and only if for all $i, j = 1,\dots, n$ $$ \frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i}.$$
Now my question is, if anybody knows some necessary or sufficient conditions for $\textbf{F}$ to be a gradient field if we replace the $C^1$ condition with a weaker condition like Lipschitz continuity or Hölder continuity (or even plane continuity if this is even possible).
I am thankful for any thoughts or possible references on this topic.