Prove for a bounded function on $[a,b]$ that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.

Problem: 

Prove for a bounded function on $[a,b]$ that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.

Answer: 

It is true that there exits a unique number $I$ such that for all partitions $P$, $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$.