MHB Is Every Function with Equal Lower and Upper Sums Constant?

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If a function f defined on the interval [a,b] has equal lower and upper sums for any partition, it can be concluded that f is a constant function. The discussion demonstrates this by considering a partition with two points, leading to the conclusion that the infimum and supremum of f over the interval must be equal. This equality implies that f takes a single constant value throughout the interval. The participants confirm the reasoning and express satisfaction with the approach. The question arises about whether other partitions could yield the same conclusion.
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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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