collection of all absolutely continuous functions on I is denoted AC(I). Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral. Differentiation The notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the "best" linear approximation. This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point , and the slope of the line is the derivative of the function at . A function is differentiable at if the limit exists. This limit is known as the derivative of at , and the function , possibly defined on only a subset of , is the derivative (or derivative function) of . If the derivative exists everywhere, the function is said to be differentiable. As a simple consequence of the definition, is continuous at if it is differentiable there. Differentiability is therefore a stronger regularity condition (condition describing the "smoothness" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see Weierstrass's nowhere differentiable continuous function). It is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on. One can classify functions by their differentiability class. The class (sometimes to indicate the interval of applicability) consists of all continuous functions. The class consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a function is exactly a function whose derivative exists and is of class . In general, the classes can be defined recursively by declaring to be the set of all continuous functions and declaring for any positive integer to be the set of all differentiable functions whose derivative is in . In particular, is contained in for every , and there are examples to show that this containment is strict. Class is the intersection of the sets as varies over the non-negative integers, and the members of this class are known as the smooth functions. Class consists of all analytic functions, and is strictly contained in (see bump function for a smooth function that is not analytic). Series A series formalizes the