Smooth homotopy theorem

Statement

1. 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 $f$
2. 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 $N$.