Is there any special significance to the term 'Real valued function" other than the fact that said function in a the standard numerical sense that assignes to real numbers from the dom F:R\to R, a real number from the co-domain. Is it taken to mean a standard function as opposed to one from sets, or one that uses complex numbers. Or does imply some kind of continuity property. Strictly montone whose range is an interval.
Or codomain is at very least connected or an closed and bounded interval of real numbers
within the prescribed interval, the best optimized number to be chosen as the solution
It means that the values that f can take on are all real.
It is to distinguish cases where f can be in the Complex domain
In the quantum theory for example the wave function is in general a complex
function with real and imaginary parts. In turn it is a function of space and time.
A representation generally means that one can find a matrix,which in the context of
matrix algera has the same algebra as the original object.
This means for example
x -y
y x
as a matrix in place of the complex number (x+iy). It is a real representation
because x and y are real.
If you just say function this is much more general than saying continuous function
all reals, or that its a function from some set of real number which outputs a real numbers. As in its just a name for a functions whose range and co-domain are not complex? Or does it mean that its defined on a real interval and has served to indicate some further structure. Such as defined on a closed and bounded domain of the reals dom(F)=[0,1] IM(F)subset [0,1] for example
You were asking about a REAL VALUED FUNCTION, which is different
from REAL VALUED ARGUMENT , You usualy say defined over.reals, or whatever.
so is generally just a name for a function whose image or co-domain is real valued.ie not complex or some other entity, so its not really anything more then what is usually considered to be a function? ie does imply extra properties like surjective or some such ?
It not imply anything like surjective. Ie it will attained all real values, Just that the values the function does attain are real valued.
Juan Weisz
a real valued function is not necessarily continuous is that right ?Again is just really a name for a typical function whose output ie image is is real valued?
The definition of a function is to assign it some unique value to every point of the domain,
over which the function is defined .
However you could assign it the value 1 if the real variable is a rational number and the value 0 if the real argument is irrational.
It would obviosly be non continuous, but any form of assignment is valid.
Thanks for @juan, i just was thinking that might have been a fit more to it, I guess it opens the door to strictly monotonic and functional or continuous/unique equations but that about it. and allows you more easily make use or derive functional addivitity due to indfferences that can be embedded onto the same domain and thus co-domain value
But in and of itself its generally just a name for a typical fuinction. Or one which has real valued outputs. I presume though, hence some of the fuss about,them in order theory there is a bit of excitement, andhey speak about a 'real valued representation by a function they presumably have in mind some of strictly monotonic real valued defined and real vlaued function from an interval to an interval, in the context of infinite order, or linear order.
A function defined from a set A to set B is a real-valued function if all the elements of the set B are real numbers. If the target space of the function f:A→B is a set of real numbers then the function f can be called a real-valued function. It can be also written as, f:A→R.
A real valued function can be defined as a function whose range or target value is of real numbers. Let us take a function f(x)=x^(1/2) where the domain is all the real numbers. The range of this function will consist of imaginary values also, and hence, it is not a real-valued function.
Each time you find the expression: "real valued function", you have to accept the fact that the function, regardless of their properties (continuous or not, convex or not, etc.), has in its range only real numbers.
Thanks for this; is there specific name for a representation by a real valued function wiih real valued domain
domain a closed and bounded interval. I presume that would be a 'real valued function 'that is in addition 'real value defined '
(ie domain and range are both real, and so is defined over a real domain, such as [0,1] and is a real valued function in that its codomain is [0,1[
Is this the same as wehn the words convex ranged is used.
For example in the context of a probability function that is strongly non atomic the words convex ranged are often used, ; ie strongly non atomic in the sense that if P(A)=x
then for all delta in [0,1] there exist event B subset A, where PR(B) =x times P(A)
presume that this just means that in some sense that if P(unit event omega)=1 one is defined, that the function is surjective and monotonic increasing for example, or continuous
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