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Showing IVP with composition of functions

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Homework Statement


Suppose that f has the intermediate value property on an interval J, that g has the intermediate value property on an interval I and that g(I) is a subset of J. Prove that f°g has intermediate value property on I.


Homework Equations





The Attempt at a Solution



I think I might have simplified too much and missed the point. Here is my work so far...


Since f has IVP on J, there is an a,b in J and a≠b and v is a number between f(a) and f(b) such that there is a c between a and b that f(c)=v.

Similarly for g, there is a e,f in I and e≠f and u is a number between g(e) and g(f) such that there is a d between e and f that g(d)=u.

Since g(I) is a subset of J, f(g(I)) is a subset of f(J).

Then f(g(e)) and f(g(f)) are numbers such that f(g(e))≠f(g(f)). Since g(d)=u is between e and f, then f(u) is between f(g(e) and f(g(f)) since f has IVP on J.

∴f°g has IVP on I.
 

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  • #2
SammyS
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Homework Statement


Suppose that f has the intermediate value property on an interval J, that g has the intermediate value property on an interval I and that g(I) is a subset of J. Prove that f°g has intermediate value property on I.

Homework Equations



The Attempt at a Solution



I think I might have simplified too much and missed the point. Here is my work so far...


Since f has IVP on J, there is an a,b in J and a≠b and v is a number between f(a) and f(b) such that there is a c between a and b that f(c)=v.

Similarly for g, there is a e,f in I and e≠f and u is a number between g(e) and g(f) such that there is a d between e and f that g(d)=u.

Since g(I) is a subset of J, f(g(I)) is a subset of f(J).

Then f(g(e)) and f(g(f)) are numbers such that f(g(e))≠f(g(f)). Since g(d)=u is between e and f, then f(u) is between f(g(e) and f(g(f)) since f has IVP on J.

∴f°g has IVP on I.
First of all -- and this is a good place to start -- What do you need to show to prove that f°g has intermediate value property on I ?
 
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First of all -- and this is a good place to start -- What do you need to show to prove that f°g has intermediate value property on I ?

Yes, that is what I am hoping to show.
 
  • #4
SammyS
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Yes, that is what I am hoping to show.
No. What I meant was:

What is it that needs to be shown in order to prove that f○g has intermediate value property on I ?

In other words: How does f○g need to behave to demonstrate that f○g has the intermediate value property on the interval, I ?
 
  • #5
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No. What I meant was:

What is it that needs to be shown in order to prove that f○g has intermediate value property on I ?

In other words: How does f○g need to behave to demonstrate that f○g has the intermediate value property on the interval, I ?
Sorry, I misread.

That for some distinct a,b in f(g(I)), for v between f(g(a)) and f(g(b)) there is a c in I that f(g(c))=v is what needs to be shown I think.
 
  • #6
SammyS
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Sorry, I misread.

That for some distinct a,b in f(g(I)), for v between f(g(a)) and f(g(b)) there is a c in I that f(g(c))=v is what needs to be shown I think.
For one thing, a & b are in I not in f(g(I)) .

For another, this must be true for every a,b in I such that a < b .

So, to go for there, for any a,b in I such that a < b, what does the fact that g has the intermediate value property on I tell you about v between g(a) and g(b) ?
 
  • #7
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For one thing, a & b are in I not in f(g(I)) .

For another, this must be true for every a,b in I such that a < b .

So, to go for there, for any a,b in I such that a < b, what does the fact that g has the intermediate value property on I tell you about v between g(a) and g(b) ?
There is a c in I so that g(c)=v. Then this extends to the composition?
 

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