Solving for Nondecreasing Continuous Functions: g(x+g(y))=g(g(x))+g(y)

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In summary: Very important...Let R denote the set of real numbers. Find all nondecreasing continuous functions g : R → R such that g(x+g(y))=g(g(x))+g(y) for all x,y∈RAnd why is that "very important"?Looks monotonous to me … :rolleyes:Looks monotonous to me … :rolleyes:I think u see it easy...I read this problem in abook,and I liked itso can you solve it?Looks monotonous to me … :rolleyes:And why is that "very important
  • #1
vip89
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Very important...

Let R denote the set of real numbers. Find all nondecreasing continuous
functions g : R R such that

g(x+g(y))=g(g(x))+g(y) for all x,y∈R
 
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  • #2
And why is that "very important"?
 
  • #3
Looks monotonous to me … :rolleyes:
 
  • #4
tiny-tim said:
Looks monotonous to me … :rolleyes:

I think u see it easy...

I read this problem in abook,and I liked it
so can you solve it?
 
  • #5
tiny-tim said:
Looks monotonous to me … :rolleyes:

HallsofIvy said:
And why is that "very important"?


I read this problem in abook,and I liked it
so can you solve it?
 
  • #6
Hi vip89! :smile:

Hint: try differentiating with respect to x, and y, separately. :smile:
 
  • #7
Oh, I see. That hint didn't seem to make much sense until I did!

(Funny how often that happens.)
 
  • #8
HallsofIvy said:
Oh, I see. That hint didn't seem to make much sense until I did!

(Funny how often that happens.)

Oh my goodness! Does it work? :smile:

I didn't actually try it! :rolleyes:
 
  • #9
...

oh...I can't solve it.!
can u send me the answer??I need it quickly and there is no time to solve it because I have final exams


Thankx very much
 
  • #10
vip89 said:
… I have final exams

Hi vip89! :smile:

It's because you have final exams that you need to practise the technique …in case it comes up!

(and if it doesn't come up … why are you bothered? :rolleyes:)

So … differentiating with respect to x … what do you get? :smile:
 
  • #11
I have acomption with my friend ,who find the answer first
 
  • #12
I am going to stay with the topic as given in the subject line here.

Why is the competition with your friend important?
 
  • #13
ok , it is not important
can u solve it ,pls??
 
  • #14
Wouldn't that be cheating?
 
  • #15
vip89 said:
ok , it is not important
can u solve it ,pls??

Actually no, I can't. But I do have some advice for you.

In this forum, you'll pretty much never get other people to solve entire problems for you. People like to see some attempt that you tried to solve it first. So far you haven't done that.

Did you try the suggestion from tiny-tim in message #6 ? If not, then try it. If you did try it then explain how far you got with it, or why you were unable to get very far with it, or just what you don't understand.

And lastly, welcome to physicsforums.com :smile:
 
  • #16
ok
I will try
and I want to say that is not cheating because It is not homework
 
  • #17
That is wt I did:
 

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What is a nondecreasing continuous function?

A nondecreasing continuous function is a type of mathematical function that always increases or stays the same as the input value increases. It is also known as a monotonic function and is represented by a graph that only moves upwards or remains flat.

How do you solve for nondecreasing continuous functions?

To solve for nondecreasing continuous functions, you need to use the functional equation g(x+g(y))=g(g(x))+g(y). This equation allows you to find the values of g(x) and g(y) by substituting different values for x and y. Once you have determined the values for g(x) and g(y), you can then plot them on a graph to visualize the function.

Are there any specific methods or techniques for solving these types of functions?

Yes, there are various methods and techniques for solving nondecreasing continuous functions. Some common approaches include using algebraic manipulation, substitution, and graphing. You can also use calculus techniques such as differentiation and integration to solve these functions.

What are the applications of nondecreasing continuous functions?

Nondecreasing continuous functions have many applications in mathematics, science, and engineering. They are commonly used to model real-world phenomena such as population growth, economic trends, and physical processes. They are also important in optimization problems, where the goal is to find the maximum or minimum value of a function.

Can nondecreasing continuous functions have more than one solution?

Yes, nondecreasing continuous functions can have multiple solutions. This is because there are often multiple combinations of g(x) and g(y) that can satisfy the functional equation g(x+g(y))=g(g(x))+g(y). However, it is important to note that some nondecreasing continuous functions may have only one unique solution.

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