sum and product and quotient of functions

Let A be a set and M a left R-module.  If  f:AM  and  g:AM,  then one may define the sum of functions f and g as the following functionMathworldPlanetmathf+g:AM:


If r is any element of the ring R, then the scalar-multiplied functionrf:AM  is defined as


Let A again be a set and K a field or a skew field.  If  f:AK  and  g:AK,  then one can define the product of functions f and g as the function  fg:AK as follows:


The quotient of functions f and g is the function  fg:{aAg(a)0}K  defined as


In particular, the incremental quotient of functions f(y)-f(x)y-x, as y tends to x, gave rise to the important concept of derivative. As another example, we can with a conscience say that the tangent ( function is the quotient of the sine ( and the cosine functions.

Title sum and product and quotient of functions
Canonical name SumAndProductAndQuotientOfFunctions
Date of creation 2013-03-22 17:44:24
Last modified on 2013-03-22 17:44:24
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Definition
Classification msc 03E20
Related topic DirectSumOfEvenoddFunctionsExample
Related topic LimitRulesOfFunctions
Related topic PolynomialFunction
Related topic ProofOfLimitRuleOfProduct
Related topic ContinuousDerivativeImpliesBoundedVariation
Related topic PropertiesOfRiemannStieltjesIntegral
Related topic InfimumAndSupremumOfSumAndProduct
Related topic PropertiesOfVectorValuedFunctio
Defines sum of functions
Defines product of functions
Defines quotient of functions
Defines scalar-multiplied function