This question was part of my analysis quiz and I couldn't solve it.
Let $f:[1/2,2] \to \mathbb{R}$ , a strictly increasing function, put $g(x)=f(x)+f(1/x) $, $x \in [1,2]$. Consider a partition $P$ of $[1,2]$ and then which of following holds:
(a) for a suitable $f$ we can have $U(P,g)=L(P,g)$
(b) For a suitable $f$ we can have $U(P,g) \neq L(P,g)$.
(c) $U(P,g) \geq L(P,g)$ for all choices of $f$.
(d) $U(p,g )<L(P,g)$ for all choices of $f $.
I have question in only (a) and (b) ie how to prove / disprove then .I need help as I was unable to do them despite trying a lot.
Thanks for help!!
Let's break this down.
Firstly, you agree that, by definition, c) is true and d) is false.
Secondly, let's examine in which cases would $U(P,g)=L(P,g)$, for a given partition $P$ of $[1,2]$. Both of these quantities are the integrals of step functions $\phi_U\geq\phi_L$ that are equal iff $g$ is a step function in $[1,2]$ that coincides with the partition $P$.
The simplest example for a $g$ that satisfies the above is $g=c$, $c\in\mathbb{R}$. Then, the question becomes can we find an $f$ st $f(x)+f(\frac{1}{x})=c$ and there is a simple example of a usual function that does that.
Finally, about any $f$ you can think of confirms b).