Given function $f$ that is continuous and defined on the closed, finite inverval $[a,b]$
Also given any two partitions $P_1$ and $P_2$ and their common refinement $P$ which consists of all of the points of $P_1$ and $P_2$
The Lower Riemann Sum is given by $L$ and the Upper Riemann Sum is given by $U$
$$L(f,P_1)\le L(f,P)\le U(f,P)\le U(f,P_2)$$
Now my calculus-book states that:
Hence, every lower sum is less than or equal to every upper sum. Since the real numbers are complete, there must exist at least one real number $I$ such that $L(f,P)\le I\le U(f,P)$ for every partition $P$. If there is only one such number, we will call it the definite integral of $f$ on $[a,b]$
Question: What are scenario's where there are indeed more than one numbers $I$ that satisfy these conditions?
I don't understand why there would be more than one number, since it is explicitly stated that $L(f,P)\le I\le U(f,P)$ for every partition $P$