sum and product and quotient of functions

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

 $(f\!+\!g)(x)\;:=\;f(x)\!+\!g(x)\quad\forall x\in A$

If $r$ is any element of the ring $R$, then the scalar-multiplied function$rf\!:A\to M$  is defined as

 $(rf)(x)\;:=\;r\!\cdot\!f(x)\quad\forall x\in A.$

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

 $(fg)(x)\;:=\;f(x)\!\cdot\!g(x)\quad\forall x\in A$

The quotient of functions $f$ and $g$ is the function  $\displaystyle\frac{f}{g}\!:\{a\in A\,\vdots\;\;g(a)\neq 0\}\to K$  defined as

 $\frac{f}{g}(x)\;:=\;\frac{f(x)}{g(x)}\quad\forall x\in A\!\smallsetminus\!\{a% \in A\,\vdots\;\;g(a)=0\}.$

In particular, the incremental quotient of functions $\frac{f(y)-f(x)}{y-x}$, 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 (http://planetmath.org/TrigonometricFunction) function is the quotient of the sine (http://planetmath.org/TrigonometricFunction) 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