Does the domain of a real valued function need be a $mathbbR$ or some subset of $mathbbR$?
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My text book says the domain and co-domain need to be either $mathbbR$ or some subset of $mathbbR$ but sometimes I read on the internet, a function that gives real values is a real valued function.
I'm not sure which one's correct though, can anyone help me out here?
For instance, would you say
$$ f , : mathbbC to mathbbR$$ is a real valued function even though the domain is a set of complex numbers?
functions
asked Aug 27 at 18:06
William
883314
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up vote
0
down vote
favorite
My text book says the domain and co-domain need to be either $mathbbR$ or some subset of $mathbbR$ but sometimes I read on the internet, a function that gives real values is a real valued function.
I'm not sure which one's correct though, can anyone help me out here?
For instance, would you say
$$ f , : mathbbC to mathbbR$$ is a real valued function even though the domain is a set of complex numbers?
functions
asked Aug 27 at 18:06
William
883314
Sometimes the term "real function" is used to denote functions whose domain and range are both subsets of $mathbbR$. The potential for confusion with "real-valued function" is of course big.
– Daniel Fischer♦
Aug 27 at 18:37
@DanielFischer hold on now, a quick google search tells me they are same? :/
– William
Aug 27 at 19:01
Depends on the author. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
– Daniel Fischer♦
Aug 27 at 19:08
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My text book says the domain and co-domain need to be either $mathbbR$ or some subset of $mathbbR$ but sometimes I read on the internet, a function that gives real values is a real valued function.
I'm not sure which one's correct though, can anyone help me out here?
For instance, would you say
$$ f , : mathbbC to mathbbR$$ is a real valued function even though the domain is a set of complex numbers?
functions
asked Aug 27 at 18:06
William
883314
My text book says the domain and co-domain need to be either $mathbbR$ or some subset of $mathbbR$ but sometimes I read on the internet, a function that gives real values is a real valued function.
I'm not sure which one's correct though, can anyone help me out here?
For instance, would you say
$$ f , : mathbbC to mathbbR$$ is a real valued function even though the domain is a set of complex numbers?
functions
asked Aug 27 at 18:06
William
883314
edited Aug 27 at 18:14
asked Aug 27 at 18:06
William
883314
asked Aug 27 at 18:06
William
883314
asked Aug 27 at 18:06
William
883314
883314
Sometimes the term "real function" is used to denote functions whose domain and range are both subsets of $mathbbR$. The potential for confusion with "real-valued function" is of course big.
– Daniel Fischer♦
Aug 27 at 18:37
@DanielFischer hold on now, a quick google search tells me they are same? :/
– William
Aug 27 at 19:01
Depends on the author. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
– Daniel Fischer♦
Aug 27 at 19:08
add a comment |Â
Sometimes the term "real function" is used to denote functions whose domain and range are both subsets of $mathbbR$. The potential for confusion with "real-valued function" is of course big.
– Daniel Fischer♦
Aug 27 at 18:37
@DanielFischer hold on now, a quick google search tells me they are same? :/
– William
Aug 27 at 19:01
Depends on the author. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
– Daniel Fischer♦
Aug 27 at 19:08
Sometimes the term "real function" is used to denote functions whose domain and range are both subsets of $mathbbR$. The potential for confusion with "real-valued function" is of course big.
– Daniel Fischer♦
Aug 27 at 18:37
Sometimes the term "real function" is used to denote functions whose domain and range are both subsets of $mathbbR$. The potential for confusion with "real-valued function" is of course big.
– Daniel Fischer♦
Aug 27 at 18:37
@DanielFischer hold on now, a quick google search tells me they are same? :/
– William
Aug 27 at 19:01
@DanielFischer hold on now, a quick google search tells me they are same? :/
– William
Aug 27 at 19:01
Depends on the author. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
– Daniel Fischer♦
Aug 27 at 19:08
Depends on the author. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
– Daniel Fischer♦
Aug 27 at 19:08
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
If $X$ is an arbitrary set, we call a function $f$ defined on $X$ real-valued so long as it maps $X$ into some subset (possibly the entirety of) the real numbers $mathbbR$.
answered Aug 27 at 18:11
WarTurtle
771518
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
1
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
add a comment |Â
up vote
1
down vote
It depends on the context. If you're taking a first course in calculus, it's usually assumed that all functions have a subset of real numbers as their domain.
But if it's a course in multivariable calculus, or complex analysis, or topology, then the domains may be subsets of the space, the complex plane, or some other arbitrary topological space.
When looking for definitions and conventions, it's better to use one reference (in this case, your textbook) and follow it. Other sources may not match it, for reasons of convention or context.
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
1
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If $X$ is an arbitrary set, we call a function $f$ defined on $X$ real-valued so long as it maps $X$ into some subset (possibly the entirety of) the real numbers $mathbbR$.
answered Aug 27 at 18:11
WarTurtle
771518
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
1
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
add a comment |Â
up vote
2
down vote
accepted
If $X$ is an arbitrary set, we call a function $f$ defined on $X$ real-valued so long as it maps $X$ into some subset (possibly the entirety of) the real numbers $mathbbR$.
answered Aug 27 at 18:11
WarTurtle
771518
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
1
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If $X$ is an arbitrary set, we call a function $f$ defined on $X$ real-valued so long as it maps $X$ into some subset (possibly the entirety of) the real numbers $mathbbR$.
answered Aug 27 at 18:11
WarTurtle
771518
If $X$ is an arbitrary set, we call a function $f$ defined on $X$ real-valued so long as it maps $X$ into some subset (possibly the entirety of) the real numbers $mathbbR$.
answered Aug 27 at 18:11
WarTurtle
771518
answered Aug 27 at 18:11
WarTurtle
771518
answered Aug 27 at 18:11
WarTurtle
771518
answered Aug 27 at 18:11
WarTurtle
771518
771518
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
1
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
add a comment |Â
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
1
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
My man you didn't answer my question. You just said exactly what I said. $X$ is a set of what? Real numbers? Complex Numbers?
– William
Aug 27 at 18:12
1
1
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William The question in your comment, "set of what?" was already answered in WarTurtle's answer: "an arbitrary set".
– Andreas Blass
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@William $X$ can be anything; it doesn't have to be a set of numbers.
– WarTurtle
Aug 27 at 18:14
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
@WarTurtle Now that I think of it, it makes sense. +1
– William
Aug 27 at 18:17
add a comment |Â
up vote
1
down vote
It depends on the context. If you're taking a first course in calculus, it's usually assumed that all functions have a subset of real numbers as their domain.
But if it's a course in multivariable calculus, or complex analysis, or topology, then the domains may be subsets of the space, the complex plane, or some other arbitrary topological space.
When looking for definitions and conventions, it's better to use one reference (in this case, your textbook) and follow it. Other sources may not match it, for reasons of convention or context.
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
1
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
add a comment |Â
up vote
1
down vote
It depends on the context. If you're taking a first course in calculus, it's usually assumed that all functions have a subset of real numbers as their domain.
But if it's a course in multivariable calculus, or complex analysis, or topology, then the domains may be subsets of the space, the complex plane, or some other arbitrary topological space.
When looking for definitions and conventions, it's better to use one reference (in this case, your textbook) and follow it. Other sources may not match it, for reasons of convention or context.
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
1
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
add a comment |Â
up vote
1
down vote
up vote
1
down vote
It depends on the context. If you're taking a first course in calculus, it's usually assumed that all functions have a subset of real numbers as their domain.
But if it's a course in multivariable calculus, or complex analysis, or topology, then the domains may be subsets of the space, the complex plane, or some other arbitrary topological space.
When looking for definitions and conventions, it's better to use one reference (in this case, your textbook) and follow it. Other sources may not match it, for reasons of convention or context.
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
It depends on the context. If you're taking a first course in calculus, it's usually assumed that all functions have a subset of real numbers as their domain.
But if it's a course in multivariable calculus, or complex analysis, or topology, then the domains may be subsets of the space, the complex plane, or some other arbitrary topological space.
When looking for definitions and conventions, it's better to use one reference (in this case, your textbook) and follow it. Other sources may not match it, for reasons of convention or context.
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
answered Aug 27 at 18:24
Matthew Leingang
15.6k12144
15.6k12144
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
1
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
add a comment |Â
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
1
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
Tell me something, why does Math depend on contexts? Isn't everything in math properly defined?
– William
Aug 27 at 18:27
1
1
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
Contexts are important because audiences vary. Students in a first year calculus course do not need to know about topological spaces to understand functions of a real variable. Everything in math is properly defined, but not all definitions agree. For instance, what is your definition of the set of natural numbers $mathbbN$? In some books you will find it defined to be the positive integers, and in others the nonnegative integers. There are good reasons for either convention.
– Matthew Leingang
Aug 27 at 18:47
add a comment |Â
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Sometimes the term "real function" is used to denote functions whose domain and range are both subsets of $mathbbR$. The potential for confusion with "real-valued function" is of course big.
– Daniel Fischer♦
Aug 27 at 18:37
@DanielFischer hold on now, a quick google search tells me they are same? :/
– William
Aug 27 at 19:01
Depends on the author. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
– Daniel Fischer♦
Aug 27 at 19:08