Increasing Function on an Interval .... Browder, Proposition 3.7 .... ....

In summary: Good luck with the rest of your studies!In summary, the conversation discusses a reader's difficulty in understanding Proposition 3.7 in Andrew Browder's "Mathematical Analysis: An Introduction." The reader asks for a formal and rigorous demonstration of A\leq f(t) \leq B, to which GJA responds by explaining how the definition of increasing functions can help resolve the issue. The reader expresses gratitude for the help and reflects on the difficulty of understanding supremums and infimums.
  • #1
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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.7 ...Proposition 3.7 and its proof read as follows:View attachment 9509
In the above proof by Andrew Browder we read the following:

" ... ... Clearly \(\displaystyle A\leq f(t) \leq B\) since \(\displaystyle f\) is increasing ... ... "
Can someone demonstrate, formally and rigorously that \(\displaystyle A\leq f(t) \leq B\) ... ...Note: Although it seems highly plausible, given the definitions of \(\displaystyle A\) and \(\displaystyle B\) and given also that \(\displaystyle f\) is increasing, that \(\displaystyle A\leq f(t) \leq B\) .. I am unable to prove it rigorously ... Hope someone can help ...

Peter
 

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Hi Peter,

Since $f$ is increasing we know $x_{1}<x_{2}\,\Longrightarrow\, f(x_{1})\leq f(x_{2}).$ This means that $f(t)$ is an upper bound for the set $\{f(s)\,\vert\, s < t\}$ and a lower bound for $\{f(s)\,\vert\, t<s\}$. Does that help resolve the issue?
 
  • #3
GJA said:
Hi Peter,

Since $f$ is increasing we know $x_{1}<x_{2}\,\Longrightarrow\, f(x_{1})\leq f(x_{2}).$ This means that $f(t)$ is an upper bound for the set $\{f(s)\,\vert\, s < t\}$ and a lower bound for $\{f(s)\,\vert\, t<s\}$. Does that help resolve the issue?
Thanks so so much for the help GJA ...

Reflecting on what you have said ...

Beginning to suspect that I'm overthinking this issue ...

Thanks again for the help ...

Peter
 
  • #4
Hi Peter,

Always happy to help in any way that I can. I had a tough time too when it came to understanding supremums and infimums. The words "least upper" and "greatest lower" don't hit the ear right initially.
 

What is Proposition 3.7 in Browder's "Increasing Function on an Interval"?

Proposition 3.7 in Browder's "Increasing Function on an Interval" states that if a function is continuous on a closed interval and its derivative is positive at each point in the interval, then the function is strictly increasing on that interval.

How can Proposition 3.7 be used to prove that a function is increasing on an interval?

Proposition 3.7 provides a necessary and sufficient condition for a function to be strictly increasing on an interval. By showing that the function is continuous on a closed interval and its derivative is positive at each point in the interval, Proposition 3.7 can be used to prove that the function is increasing on that interval.

Why is it important to study increasing functions on an interval?

Studying increasing functions on an interval is important because it allows us to understand the behavior of a function and make predictions about its values. It also helps us to identify critical points and intervals where the function is increasing or decreasing, which can be useful in optimization problems.

What are some real-life applications of Proposition 3.7 and increasing functions on an interval?

Proposition 3.7 and increasing functions on an interval have many real-life applications, such as in economics, where they can be used to model and analyze the behavior of demand and supply curves. They are also useful in physics, where they can be used to study the motion of objects and the rate of change of physical quantities over time.

Are there any limitations to Proposition 3.7 and the concept of increasing functions on an interval?

One limitation of Proposition 3.7 is that it only applies to continuous functions on closed intervals. It cannot be used to prove that a function is increasing on an open interval or a non-continuous function. Additionally, the concept of increasing functions on an interval may not be applicable to all types of functions, such as discontinuous or multi-valued functions.

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