I'm trying to understand the hypothesis upon the definition of curl and divergence of a function $f$. Why is $f$ needed continuously differentiable? From Wikipedia:
As such, the curl operator maps continuously differentiable functions $f:ℝ^3 → ℝ^3$ to continuous functions $g: ℝ^3 → ℝ^3$.
In fact, it maps $C^k$ functions in $\mathbb{R}^3$ to $C^{k-1}$ functions in $\mathbb{R}^3$.
Is the $C^1$ condition a necessary one?
In Rudin's Principle of Mathematical Analysis:
Suppose now $F$ is a vector field defined over an open subset $E\subset\mathbb{R}^3$, of class $C^1$. Then its curl and divergence are defined by ...
Since the $C^1$ condition is not needed for gradient, I think it is related to the fact that both curl and divergence contains derivatives with respect to multiple variables in the same component (the only one for divergence, anyone for the curl). Am I on the right path?