Let $M \subset \mathbb{R}^N$ be a smooth manifold. Let $f:\mathbb{R}^N \to \mathbb{R}$ be an analytic function.
I wonder if the restriction $f|_M$ is analytic?
My motivation to write this question is to understand the embedding theorem from Whitney which states that a differentiable manifold can be smoothly embedded in $\mathbb{R}$. Hence some people claims (without proof ) that $M$ admits an analytic structure.