A question involving self-composition.

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In summary, the conversation discusses a problem involving a differentiable function and its composition. The person has found a proof for the problem but is unsure if it is complete. They also mention investigating fixed points of the function, but have not found a solution yet. The other person suggests that if the function is monotone decreasing but not strictly decreasing, then it must be constant. The first person asks for further explanation about this standard result.
  • #1
zpconn
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I can't for the life of me figure this one out all the way.

Suppose f : R -> R is differentiable, and consider g(x) = f(f(x)). Show that if g is monotone decreasing, then g must be constant.

Here's what I've done so far (I'd hesitate to call it "progress"):

By the chain rule, g'(x) = f'(f(x)) f'(x). Suppose that g is strictly decreasing so that f'(f(x)) f'(x) < 0. One of the factors is positive and the other is negative. Since, by Darboux's theorem, f'(x) has the intermediate value property, there exists a q between x and f(x) such that f'(q) = 0. Then g'(q) = f'(f(q)) f'(q) = 0, a contradiction. Therefore g is not strictly decreasing.

I've investigated the fixed points of f and found lots of interesting facts, but none of them seems to lead anywhere on this problem.

Any ideas or suggestions? Thanks a lot.
 
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  • #2
You have your proof already. "Strictly decreasing" and "monotone decreasing" mean two different things. If g is monotone decreasing but not strictly decreasing, then g must be constant.
 
  • #3
Why is that?

If g is monotone decreasing and not strictly decreasing, than means nothing more than that there's at least one u at which g'(u) = 0, i.e. g has at least one inflection point. However, if g were constant, it'd have g'(x) = 0 everywhere. Is there some standard result about monotone but not strictly decreasing functions that I'm unaware of?
 
Last edited:

What is self-composition?

Self-composition refers to the process of creating something (such as a piece of music, a written work, or a piece of art) by using elements of one's own previous work.

Why is self-composition important?

Self-composition allows for the creation of unique and original works that reflect an individual's personal experiences and ideas. It also allows for the development and refinement of artistic skills and techniques.

What are some examples of self-composition?

Examples of self-composition include an artist creating a series of paintings based on their own previous works, a musician sampling their own music to create a new song, or a writer incorporating elements and characters from their previous stories into a new novel.

What are the benefits of self-composition?

Self-composition can help artists to develop a distinct and recognizable style, as well as provide a sense of creative satisfaction and fulfillment. It also allows for the exploration and expression of personal ideas and emotions.

How can self-composition be used in scientific research?

In scientific research, self-composition can involve building upon previous studies and experiments to create new and innovative methods or theories. It can also involve using data and findings from previous research to support or challenge current hypotheses or conclusions.

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