- #1
evinda
Gold Member
MHB
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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 $L(f,P)=U(f,P) \Rightarrow inf([a,b])=supf([a,b])=c \Rightarrow f(x)=c$
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 $L(f,P)=U(f,P) \Rightarrow inf([a,b])=supf([a,b])=c \Rightarrow f(x)=c$
