F is strictly increasing at each point in (a,b)

In summary, the challenge problem is to prove that a function is strictly increasing on a bounded open interval if it is strictly increasing at each point in the interval. This is done by considering the definition of strictly increasing at a point and using properties such as compactness to simplify the proof.
  • #1
Mr Davis 97
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Homework Statement


Let ##(a,b)\subset \mathbb{R}## be a bounded open interval. Prove that if ##f:(a,b)\to\ \mathbb{R}## is strictly increasing at each point in ##(a,b)##, then ##f## is strictly increasing on ##(a,b)##.

Homework Equations


Let ##(a,b)\subset \mathbb{R}## be a bounded open interval in ##\mathbb{R}##. We say that ##f:(a,b)\to \mathbb{R}## is strictly increasing at ##c\in(a,b)## if there is a ##\delta>0## such that both the following hold:
1) for each ##x\in(c-\delta,c)## we have ##f(x)<f(c)##
2) for each ##x\in(c,c+\delta)## we have ##f(c)<f(x)##.

We say that ##f:(a,b)\to \mathbb{R}## is strictly increasing on ##(a,b)## if whenever ##x,y\in(a,b)## with ##x<y##, we find that ##f(x)<f(y)##.

The Attempt at a Solution



I am not sure where to start here. This is a challenge problem in the section on the least upper bound, so I suppose it relates somehow, but I am not seeing the connection. Hints would be nice.

EDIT: I think I solved it.
 
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  • #2
Mr Davis 97 said:
1) for each ##x\in(c-\delta,c)## we have ##f(x)<f(c)##
2) for each ##x\in(c,c+\delta)## we have ##f(c)<f(x)##

The way I did it, I combined these two statements, changed the second ##x## to a ##y## and chose a ##\delta>0## such that ##(c-\delta,c+\delta)\subset (a,b)##.
 
  • #3
Eclair_de_XII said:
The way I did it, I combined these two statements, changed the second ##x## to a ##y## and chose a ##\delta>0## such that ##(c-\delta,c+\delta)\subset (a,b)##.
I don't think you're allowed to choose ##\delta##. It says only that "there exists."
 
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  • #4
did they prove the heine borel property in the section on lub's? (i.e. every open covering of a closed bounded interval by open intyervals has a finite sub covering.) given points c< d in the interval, the fact that f(c) < f(d) seems to follow from the existence of a finite cover of the closed bounded interval [c,d] by open intervals of the type mentioned.

or did they discuss greatest lower bounds? given c in (a,b) you want to show there are no points x >c where f(x) ≤ f(c). Assuming there are some, the set of such x is a bounded non empty set and has a greatest lower bound d. That means every x with f(x) ≤ f(c) has x ≥ d, and also that there are such x arbitrarily close to d and greater than d.

case 1) f(d) ≤ f(c). rule this out.

case 2) f(d) > f(c). Rule this out.
 
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  • #5
Mr Davis 97 said:

Homework Statement


Let ##(a,b)\subset \mathbb{R}## be a bounded open interval. Prove that if ##f:(a,b)\to\ \mathbb{R}## is strictly increasing at each point in ##(a,b)##, then ##f## is strictly increasing on ##(a,b)##.

Homework Equations


Let ##(a,b)\subset \mathbb{R}## be a bounded open interval in ##\mathbb{R}##. We say that ##f:(a,b)\to \mathbb{R}## is strictly increasing at ##c\in(a,b)## if there is a ##\delta>0## such that both the following hold:
1) for each ##x\in(c-\delta,c)## we have ##f(x)<f(c)##
2) for each ##x\in(c,c+\delta)## we have ##f(c)<f(x)##.

We say that ##f:(a,b)\to \mathbb{R}## is strictly increasing on ##(a,b)## if whenever ##x,y\in(a,b)## with ##x<y##, we find that ##f(x)<f(y)##.

The Attempt at a Solution



I am not sure where to start here. This is a challenge problem in the section on the least upper bound, so I suppose it relates somehow, but I am not seeing the connection. Hints would be nice.

EDIT: I think I solved it.
What is the definition of the concept "is strictly increasing at ##c \in (a,b)##"? Using some definitions the problem is trivial, but using others it presents a bit of a challenge.
 
  • #6
I think this may work: choose x>y. Then there is a chain of ##\delta##'s from y to x, but maybe you can join them with a line that includes the endpoints and then use compactness to use only finitely-many ##\delta##s.
 

1. What does it mean for F to be strictly increasing?

It means that as the input values of the function increase, the output values also increase without any decrease in between.

2. How can we determine if F is strictly increasing?

We can determine this by looking at the slope of the function. If the slope is positive at every point in the given interval (a,b), then the function is strictly increasing.

3. What is the difference between strictly increasing and just increasing?

Strictly increasing means that the function is increasing without any decrease in between, while increasing allows for the possibility of flat or constant segments within the interval.

4. What are some examples of functions that are strictly increasing?

Linear, exponential, and power functions are all examples of functions that are strictly increasing. In general, any function with a positive slope will be strictly increasing.

5. Can a function be strictly increasing at one point and not at others?

Yes, it is possible for a function to be strictly increasing at one point and not at others. This can occur if the function has a sharp increase or decrease at a specific point, but is otherwise increasing or decreasing.

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