Obviously, there are continuous maps that are not smooth.
The question is, can you approximate any continuous function arbitrarily close using a smooth one.
The answer: yes!
Statement (for functions to )
Suppose is a smooth manifold and is a continuous function.
For any positive continuous function , there exists a smooth function that is close to .
If is smooth on a closed subset , then can be chosen such that .
For smooth manifolds ( without boundary) and a continuous map, is homotopic to a smooth map.
If is already smooth on a closed subset , then the homotopy can be taken to be relative to .