The various order of parametric continuity can be described as follows:
- C−1: Curves are discontinuous
- C0: Curves are continuous
- C1: First derivatives are continuous
- C2: First and second derivatives are continuous
- Cn: First through nth derivatives are continuous
Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer. The function f is said to be of (differentiability) class Ckif the derivatives f′, f′′, …, f(k) exist and are continuous (the continuity is implied by differentiability for all the derivatives except for f(k)).
To put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k, and there are examples to show that this containment is strict. C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers (i.e. from 0 to ∞).
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 have continuous derivatives.
functions are also called “smooth” because neither they nor their derivatives have “corners,” which would make their graph look somewhat rough. For example, is not smooth (right figure above).