Editing Function of a real variable (section)

===Basic examples===

For many commonly used real functions, the domain is the whole set of real numbers, and the function is [[continuous function|continuous]] and [[differentiable function|differentiable]] at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:
* All [[polynomial function]]s, including [[constant function]]s and [[linear function (calculus)|linear functions]]
* [[Sine]] and [[cosine]] functions
* [[Exponential function]]

Some functions are defined everywhere, but not continuous at some points. For example
* The [[Heaviside step function]] is defined everywhere, but not continuous at zero.

Some functions are defined and continuous everywhere, but not everywhere differentiable. For example
* The [[absolute value]] is defined and continuous everywhere, and is differentiable everywhere, except for zero.
* The [[cubic root]] is defined and continuous everywhere, and is differentiable everywhere, except for zero.

Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:
* A [[rational function]] is a quotient of two polynomial functions, and is not defined at the [[zero of a function|zeros]] of the denominator.
* The [[tangent function]] is not defined for <math>\frac\pi 2 + k\pi,</math> where {{mvar|k}} is any integer.
* The [[logarithm function]] is defined only for positive values of the variable.

Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of:
*The [[square root]] is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).