I have found many textbooks that state the method of Lagrange multipliers as follows:
Theorem. Let $f, g: \mathbb R^n \to \mathbb R$ be $C^1$ functions and $S=\{x \in \mathbb R^n \ | \ g(x)=0\}$ be a surface such that $\nabla g(x) \neq 0$ for all $x \in S$. Then, if $f|_S$ has an extremum at $x_0$, there exists a $\lambda \in \mathbb R$ such that $\nabla f(x_0) = \lambda \nabla g(x_0)$.
Theorem. Let $f, g_1, \dotsc, g_m: \mathbb R^n \to \mathbb R$ be $C^1$ functions and $S=\{x \in \mathbb R^n \ | \ g_1(x)=\cdots=g_m(x)=0\}$ be a surface such that $\{\nabla g_1(x), \dotsc, \nabla g_m(x)\}$ are linearly independent for all $x \in S$. Then, if $f|_S$ has an extremum at $x_0$, the gradient $\nabla f(x_0)$ is in the linear span of the vectors $\{\nabla g_1(x_0), \dotsc, \nabla g_m(x_0)\}$.
The requirement that the constraint functions be $C^1$ is necessary to apply the Implicit Function Theorem.
However, despite the fact that nearly all resources I have read require the function $f$ to be $C^1$, none of the proofs appear to use this assumption. Do the theorems hold with the weaker assumption that $f$ is merely differentiable?