Smooth homotopy theorem

Statement

1. Suppose $f:M\to N$ is a continuous map between differential manifolds. Then, there exists a smooth map $f^{\prime }:M\to N$ such that $f^{\prime }$ 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 $F^{\prime }:M\times I\to N$ from $f_0$ to $f_1$ that is smooth as a map from $M\times I$ to $N$.