Is Every Function with Equal Lower and Upper Sums Constant?

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Discussion Overview

The discussion centers around the question of whether a function \( f: [a,b] \to \mathbb{R} \) that has equal lower and upper sums for any partition must necessarily be a constant function. The scope includes mathematical reasoning and exploration of the properties of functions in relation to partitions.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant proposes that if \( L(f,P) = U(f,P) \) for any partition \( P \), then the function \( f \) must be constant, using a specific partition of two points to illustrate this.
  • Several participants express agreement with the initial proposal, affirming that the reasoning appears correct.
  • Another participant questions whether the chosen partition is the only one that could be used to demonstrate the claim.

Areas of Agreement / Disagreement

Participants generally agree with the initial reasoning presented, but there is a question raised about the uniqueness of the partition used, indicating that some uncertainty remains regarding the completeness of the argument.

Contextual Notes

The discussion does not resolve whether other partitions could lead to different conclusions or if the argument holds universally across all possible partitions.

evinda
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Hello! :rolleyes:

Let $f:[a,b] \to \mathbb{R}$ function with the identity:
$$L(f,P)=U(f,P), \text{ for any partition } P \text{ of } [a,b]$$

Show that $f$ is a constant.

Can I do it like that?

We pick for the interval $[a,b]$ a partition of $2$ points: $P=\{a=x_0,x_1=b\}$

Then,it is like that: $L(f,P)=(b-a)inf([a,b]) \text{ and } U(f,P)=(b-a)supf([a,b])$

Since we know that $L(f,P)=U(f,P) \Rightarrow inf([a,b])=supf([a,b])=c \Rightarrow f(x)=c$
 
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Yes, that looks good. (Yes)
 
Opalg said:
Yes, that looks good. (Yes)

Nice..Thank you! :)
 
Opalg said:
Yes, that looks good. (Yes)

Is this partition the only possible that I could take? (Thinking)
 

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