A C^∞ function is a function that is differentiable for all degrees of differentiation. For instance, f(x) = e^(2x) (left figure above) is C^∞ because its nth derivative f^(n)(x) = 2^n e^(2x) exists and is continuous. All polynomials are C^∞. The reason for the notation is that C^k have k continuous derivatives. C^∞ functions are also called "smooth" because neither they nor their derivatives have "corners, " which would make their graph look somewhat rough. For example, f(x) = left bracketing bar x^3 right bracketing bar is not smooth (right figure above).

calculus | C^k function | differential topology | Fréchet space | partition of unity | Sard's theorem