The various order of parametric continuity can be described as follows:^{}

*C*^{−1}: Curves are discontinuous*C*^{0}: Curves are continuous*C*^{1}: First derivatives are continuous*C*^{2}: First and second derivatives are continuous*C*: First through^{n}*n*th 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 C^{k}**if 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 *C*^{0} consists of all continuous functions. The class *C*^{1} consists of all differentiable functions whose derivative is continuous; such functions are called **continuously differentiable**. Thus, a *C*^{1} function is exactly a function whose derivative exists and is of class *C*^{0}. In general, the classes *C ^{k}* can be defined recursively by declaring

*C*

^{0}to be the set of all continuous functions and declaring

*C*for any positive integer

^{k}*k*to be the set of all differentiable functions whose derivative is in

*C*

^{k−1}. In particular,

*C*is contained in

^{k}*C*

^{k−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

*C*as

^{k}*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).

https://en.wikipedia.org/wiki/Smoothness