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

[evinda]

Hello!

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 $L(f,P)=U(f,P) \Rightarrow inf([a,b])=supf([a,b])=c \Rightarrow f(x)=c$

 

[Opalg]

Yes, that looks good. (Yes)

 

[evinda]

Yes, that looks good. (Yes)

Nice..Thank you! :)

 

[evinda]

Yes, that looks good. (Yes)

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