Is Every Function with Equal Lower and Upper Sums Constant?

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SUMMARY

The discussion confirms that if a function \( f: [a,b] \to \mathbb{R} \) satisfies the condition \( L(f,P) = U(f,P) \) for any partition \( P \) of the interval \([a,b]\), then \( f \) must be a constant function. The proof utilizes a partition with two points, demonstrating that the infimum and supremum of \( f \) over \([a,b]\) must be equal, leading to the conclusion that \( f(x) = c \) for some constant \( c \). The participants also inquire about the uniqueness of the partition used in the proof.

PREREQUISITES
  • Understanding of lower and upper sums in the context of Riemann integration
  • Familiarity with the concepts of infimum and supremum
  • Basic knowledge of function properties in real analysis
  • Experience with partitioning intervals in mathematical proofs
NEXT STEPS
  • Study the properties of Riemann integrable functions
  • Explore the implications of constant functions in real analysis
  • Learn about different types of partitions and their effects on integration
  • Investigate the relationship between continuity and constancy of functions
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Mathematics students, educators, and anyone interested in real analysis, particularly those studying properties of functions and Riemann integration.

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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