Does the sum of two functions satisfying the intermediate value property also have this property?
up vote
5
down vote
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If functions $f$ and $g$ both satisfy the intermediate value property, does their sum also satisfy this property? If not, what if I suppose in addition that $f$ is continuous?
Thanks in advance!
Edit: I found the second part of my question here:
Is the sum of a Darboux function and a continuous function Darboux?
calculus analysis continuity
edited 24 mins ago
Glorfindel
3,38481730
asked 1 hour ago
Jiu
41619
add a comment |
up vote
5
down vote
favorite
If functions $f$ and $g$ both satisfy the intermediate value property, does their sum also satisfy this property? If not, what if I suppose in addition that $f$ is continuous?
Thanks in advance!
Edit: I found the second part of my question here:
Is the sum of a Darboux function and a continuous function Darboux?
calculus analysis continuity
edited 24 mins ago
Glorfindel
3,38481730
asked 1 hour ago
Jiu
41619
The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order)
– Yanko
59 mins ago
add a comment |
up vote
5
down vote
favorite
up vote
5
down vote
favorite
If functions $f$ and $g$ both satisfy the intermediate value property, does their sum also satisfy this property? If not, what if I suppose in addition that $f$ is continuous?
Thanks in advance!
Edit: I found the second part of my question here:
Is the sum of a Darboux function and a continuous function Darboux?
calculus analysis continuity
edited 24 mins ago
Glorfindel
3,38481730
asked 1 hour ago
Jiu
41619
If functions $f$ and $g$ both satisfy the intermediate value property, does their sum also satisfy this property? If not, what if I suppose in addition that $f$ is continuous?
Thanks in advance!
Edit: I found the second part of my question here:
Is the sum of a Darboux function and a continuous function Darboux?
calculus analysis continuity
calculus analysis continuity
edited 24 mins ago
Glorfindel
3,38481730
asked 1 hour ago
Jiu
41619
edited 24 mins ago
Glorfindel
3,38481730
asked 1 hour ago
Jiu
41619
edited 24 mins ago
Glorfindel
3,38481730
edited 24 mins ago
Glorfindel
3,38481730
edited 24 mins ago
Glorfindel
3,38481730
3,38481730
asked 1 hour ago
Jiu
41619
asked 1 hour ago
Jiu
41619
asked 1 hour ago
Jiu
41619
41619
The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order)
– Yanko
59 mins ago
add a comment |
The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order)
– Yanko
59 mins ago
The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order)
– Yanko
59 mins ago
The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order)
– Yanko
59 mins ago
add a comment |
1 Answer
1
active
oldest
votes
up vote
7
down vote
accepted
Consider the functions, $f:[0,1]rightarrow [0,1]$ and $g:[0,1]rightarrow[0,1]$ where
$$f = begin{cases}sinfrac{1}{x},& x>0 \ 0, & x = 0end{cases}$$
and
$$g = begin{cases}-sinfrac{1}{x},& x>0 \ 1, & x = 0end{cases}$$
answered 1 hour ago
Olof Rubin
771213
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
Consider the functions, $f:[0,1]rightarrow [0,1]$ and $g:[0,1]rightarrow[0,1]$ where
$$f = begin{cases}sinfrac{1}{x},& x>0 \ 0, & x = 0end{cases}$$
and
$$g = begin{cases}-sinfrac{1}{x},& x>0 \ 1, & x = 0end{cases}$$
answered 1 hour ago
Olof Rubin
771213
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
add a comment |
up vote
7
down vote
accepted
Consider the functions, $f:[0,1]rightarrow [0,1]$ and $g:[0,1]rightarrow[0,1]$ where
$$f = begin{cases}sinfrac{1}{x},& x>0 \ 0, & x = 0end{cases}$$
and
$$g = begin{cases}-sinfrac{1}{x},& x>0 \ 1, & x = 0end{cases}$$
answered 1 hour ago
Olof Rubin
771213
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
add a comment |
up vote
7
down vote
accepted
up vote
7
down vote
accepted
Consider the functions, $f:[0,1]rightarrow [0,1]$ and $g:[0,1]rightarrow[0,1]$ where
$$f = begin{cases}sinfrac{1}{x},& x>0 \ 0, & x = 0end{cases}$$
and
$$g = begin{cases}-sinfrac{1}{x},& x>0 \ 1, & x = 0end{cases}$$
answered 1 hour ago
Olof Rubin
771213
Consider the functions, $f:[0,1]rightarrow [0,1]$ and $g:[0,1]rightarrow[0,1]$ where
$$f = begin{cases}sinfrac{1}{x},& x>0 \ 0, & x = 0end{cases}$$
and
$$g = begin{cases}-sinfrac{1}{x},& x>0 \ 1, & x = 0end{cases}$$
answered 1 hour ago
Olof Rubin
771213
answered 1 hour ago
Olof Rubin
771213
answered 1 hour ago
Olof Rubin
771213
answered 1 hour ago
Olof Rubin
771213
771213
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
add a comment |
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
Thanks for your answer! What if we suppose that one of the functions is continuous? Or should I ask this in another question?
– Jiu
51 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I'm not sure whether it is true or not in that case. Sorry
– Olof Rubin
38 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
I just found the same question, see my edit!
– Jiu
27 mins ago
add a comment |
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The best thing you can aim for is to show that the image of $f+g$ is an interval. (In particular it takes all the values in-between but not necessarily in the right order)
– Yanko
59 mins ago