Editing Continuous function (section)

===Rules for continuity===
[[File:Brent method example.svg|right|thumb|The graph of a [[cubic function]] has no jumps or holes. The function is continuous.]]

Proving the continuity of a function by a direct application of the definition is generaly a noneasy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules:

* Every [[constant function]] is continuous
* The [[identity function]] {{tmath|1=f(x) = x}} is continuous
* ''Addition and multiplication:'' If the functions {{tmath|f}} and {{tmath|g}} are continuous on their respective domains {{tmath|D_f}} and {{tmath|D_g}}, then their sum {{tmath|f+g}} and their product {{tmath|f\cdot g}} are continuous on the [[set intersection|intersection]] {{tmath|D_f\cap D_g}}, where {{tmath|f+g}} and {{tmath|fg}} are defined by {{tmath|1=(f+g)(x)=f(x)+g(x)}} and {{tmath|1=(f\cdot g)(x)=f(x)\cdot g(x)}}.
* ''[[Multiplicative inverse|Reciprocal]]:'' If the function {{tmath|f}} is continuous on the domain {{tmath|D_f}}, then its reciprocal {{tmath|\tfrac 1 f}}, defined by {{tmath|1=(\tfrac 1  f)(x)= \tfrac 1{f(x)} }} is continuous on the domain {{tmath|1=D_f\setminus f^{-1}(0)}}, that is, the domain {{tmath|D_f}} from which the points {{tmath|x}} such that {{tmath|1=f(x)=0}} are removed.
* ''[[Function composition]]:'' If the functions {{tmath|f}} and {{tmath|g}} are continuous on their respective domains {{tmath|D_f}} and {{tmath|D_g}}, then the composition {{tmath|g\circ f}} defined by {{tmath|(g\circ f)(x) = g(f(x))}} is continuous on {{tmath|D_f\cap f^{-1}(D_g)}}, that the part of {{tmath|D_f}} that is mapped by {{tmath|f}} inside {{tmath|D_g}}. 
* The [[sine and cosine]] functions ({{tmath|\sin x}} and {{tmath|\cos x}}) are continuous everywhere.
* The [[exponential function]] {{tmath|e^x}} is continuous everywhere.
* The [[natural logarithm]] {{tmath|\ln x}} is continuous on the domain formed by all positive real numbers {{tmath|\{x\mid x>0\} }}.

[[File:Homografia.svg|right|thumb|The graph of a continuous [[rational function]]. The function is not defined for <math>x = -2.</math> The vertical and horizontal lines are [[asymptote]]s.]]

These rules imply that every [[polynomial function]] is continuous everywhere and that a [[rational function]] is continuous everywhere where it is defined, if the numerator and the denominator have no common [[zero of a function|zeros]]. More generally, the quotient of two continuous functions is continuous outside the zeros of the denominator.

[[File:Si cos.svg|thumb|The sinc and the cos functions]]

An example of a function for which the above rules are not sufficirent is the [[sinc function]], which is defined by {{tmath|1=\operatorname{sinc}(0)=1 }} and {{tmath|1=\operatorname{sinc}(x)=\tfrac{\sin x}{x} }} for {{tmath|x\neq 0}}. The above rules show immediately that the function is continuous for {{tmath|x\neq 0}}, but, for proving the continuity at {{tmath|0}}, one has to prove 
<math display="block">\lim_{x\to 0} \frac{\sin x}{x} = 1.</math>
As this is true, one gets that the sinc function is continuous function on all real numbers.