Editing Real analysis (section)

===Differentiation===
{{Main|Derivative|Differential calculus|}}
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 <math>a</math>, and the slope of the line is the derivative of the function at <math>a</math>.

A function <math>f:\mathbb{R}\to\mathbb{R}</math> is '''''differentiable at <math>a</math>''''' if the [[limit of a function|limit]]

:<math>f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}</math>

exists.  This limit is known as the '''''derivative of <math>f</math> at <math>a</math>''''', and the function <math>f'</math>, possibly defined on only a subset of <math>\mathbb{R}</math>, is the '''''derivative''''' (or '''''derivative function''''') '''''of''''' '''''<math>f</math>'''''.  If the derivative exists everywhere, the function is said to be '''''differentiable'''''.

As a simple consequence of the definition, <math>f</math> is continuous at '''''<math>a</math>''''' 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 function|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 <math>C^0</math> (sometimes <math>C^0([a,b])</math> to indicate the interval of applicability) consists of all continuous functions.  The class <math>C^1</math> consists of all [[differentiable function]]s whose derivative is continuous; such functions are called '''''continuously differentiable'''''.  Thus, a <math>C^1</math> function is exactly a function whose derivative exists and is of class <math>C^0</math>.  In general, the classes ''<math>C^k</math>'' can be defined [[recursion|recursively]] by declaring <math>C^0</math> to be the set of all continuous functions and declaring ''<math>C^k</math>'' for any positive integer <math>k</math> to be the set of all differentiable functions whose derivative is in <math>C^{k-1}</math>.  In particular, ''<math>C^k</math>'' is contained in <math>C^{k-1}</math> for every <math>k</math>, and there are examples to show that this containment is strict.  Class <math>C^\infty</math> is the intersection of the sets ''<math>C^k</math>'' as ''<math>k</math>'' varies over the non-negative integers, and the members of this class are known as the '''''smooth functions'''''.  Class <math>C^\omega</math> consists of all [[analytic function]]s, and is strictly contained in <math>C^\infty</math> (see [[bump function]] for a smooth function that is not analytic).