C^infty Function

A function is a function that is differentiable for all degrees of differentiation. For instance, (left figure above) is because its th derivative exists and is continuous. All polynomials are . The reason for the notation is that C^k functions have continuous derivatives.

functions are also called "smooth" because neither they nor their derivatives have "corners," which would make their function graph look somewhat rough. For example, is not smooth (right figure above).

There are special functions which are very useful in analysis and geometry. For example, there are smooth functions called bump functions, which are smooth approximations to a characteristic function. Typically, these functions require some calculus to show that they are indeed .

Any real analytic function is smooth. But a smooth function is not necessarily real analytic. For instance, a nonzero real analytic function on cannot be a bump function. Consider the following function, whose Taylor series at 0 is identically zero, yet the function is not zero:

(1)

The function goes to zero very quickly as . One property of smooth functions is that they can look very different at different scales.

The function space of smooth functions on a nondegenerate compact closed interval, with the topology of uniform convergence of every derivative, is a Fréchet space. This topology cannot be defined by a single norm, so the function space is not a Banach space in this topology.