Definition.
An $m$-dimensional submanifold of $\mathbb{R}^n$ is a subset $M\subset\mathbb{R}^n$ s.t. for all $x\in M$, there exists a neighbourhood of $x$, say $U$, an open set $V\subset\mathbb{R}^n$ and a diffeomorphism $\varphi$ where $\varphi(U\cap M) = V\cap(\mathbb{R}^m\times 0)$ .
If I am not mistaken, I look through a proof where diffeomorphism assumed to be continuously differentiable to use inverse function theorem.
Do we require diffeomorphism to be continuously differentiable in the definition of submanifolds?
I appreciate if someone can elaborate the difference.