Smooth homotopy theorem

Statement

Suppose $f : M \to N$ is a continuous map between differential manifolds. Then, there exists a smooth map $f^{'} : M \to N$ such that $f^{'}$ is homotopy-equivalent to $f$
Suppose $F : M \times I \to N$ is a homotopy between smooth maps $f_{0} : M \to N$ and $f_{1} : M \to N$ . Then, there exists a homotopy $F^{'} : M \times I \to N$ from $f_{0}$ to $f_{1}$ that is smooth as a map from $M \times I$ to $N$ .