Upper/lower sum definition (spivak)

Question

The first (and only) answer to this question: Defintion of Upper & Lower Riemann Sum reminds us of the definition of Upper and lower Riemann sums. My question is about the usage of infimum and supremum for each of the subintervals. Can't these terms be replaced my minimum and maximum? Why not? Is there an example of a function defined and bounded in a closed space whose infimum is not its minimum?

Answer 1

Yes. For a somewhat contrived example, consider the function $$ f(x) = \begin{cases} \vert x \vert & \text{ if } x \neq 0 \\ 1 & \text{ if } x = 0 \end{cases}.$$ This is clearly integrable, even on the intervals $[-1,1]$ or $[0,1]$. But this function does not have a minimum value at all.

There is no reason to exclude these functions from consideration for integration. In a slightly higher sense, this also gives intuition to two big theorems about integration.

Firstly, such bad behaviors never happen for continuous functions, and continuous functions are integrable. Each discontinuity gives another problem point. If there are too many discontinuities, then these infima and suprema pose too many problems. In fact, countably many discontinuities are okay, but uncountably many are too much.

Answer 2

No, you can't replace it.
Take $f(x) = x \quad \forall x \in [0,1]$
Any lower sum of every partition of $[0,1]$ will be strictly smaller than the infimum $1/2$.
The reason is that the partition has to be finite.
So you will always lose some 'area' for each fixed partition.
Only a limit can achieve the exact value.