The question has two part,
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Show that if f : [a, b] → R is continuous and there exists a partition P of [a,b] such that U(f,P) = L(f,P), then f is constant.
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Is this true if we drop the assumption that f is continuous?
I've already finished the proof of first question by using the Intermediate Value Theorem.(since the sup and inf of a continuous function are equal on each subinterval, then the function is constant on each subinterval. With the assumption that f is continuous, then we can conclude that f is constant in its domain.)
As for the second question, it seems to be false since if f is not continuous, the Mean Value theorem cannot be applied. However, I cannot think of a counter example or proof for this. Could anyone give a hint?
Best Answer
Let $f$ be Riemann integrable and that there is a parition $P : 0= x_0 < x_1 < \cdots <x_n = 1$ so that $U(f, P) = L(f, P)$.
This mean that for all $i = 1, \cdots n$, we have
$$\sup_{x\in [x_{i-1}, x_i]} f(x) = \inf_{x\in [x_{i-1}, x_i]} f(x)$$
That is, $f$ is constant on $[x_{i-1}, x_i]$. As the intervals contains the end points, one can show that $f$ is constant. Thus it has nothing to do with continuity of $f$.