Smoothness classes

Summary

The smoothness classes $C^0, C^1, C^2, \ldots, C^\infty$ sort functions by how many continuous derivatives they have. A function of class $C^k$ has $k$ derivatives, and the $k$-th one is continuous. The classes are nested, each strictly inside the one before, and their intersection is the class $C^\infty$ of smooth functions. Theorems in analysis and geometry state their hypotheses in terms of these classes.