Smoothness

A function $f_i$ from a subset of ${\mathbb{R}}^n$ into ${\mathbb{R}}$ is called a smooth function if derivatives of any order can be taken with respect to any variables, at any point in the domain of $f_i$. A vector field is said to be smooth if every one of its $n$ defining functions, $f_1$, $\ldots$, $f_n$, is smooth. An alternative name for a smooth function is a $C^\infty$ function. The superscript represents the order of differentiation that can be taken. For a $C^k$ function, its derivatives can be taken at least up to order $k$. A $C^0$ function is an alternative name for a continuous function. The notion of a homeomorphism can be extended to a diffeomorphism, which is a homeomorphism that is a smooth function. Two topological spaces are called diffeomorphic if there exists a diffeomorphism between them.