Assumptions for gradient to be perpendicular to level set

Question

The fact that a differentiable function is perpendicular to its level sets has been asked before. See: Gradient is perpendicular to level set and implicit function theorem and Why is the gradient always perpendicular to level curves?. Underlying every proof that I have seen of this fact is the assumption that the implicit function theorem can be applied. However, the implicit function theorem requires continuous differentiability. Is it the case then that this fact holds only under the assumption that the function be continuously differentiable, rather than just differentiable?