Let $f$ and $g$ be Riemann integrable (real) functions and
$f(x)\leq h(x)\leq g(x).$
Is it true that $h(x)$ is Riemann integrable? Can someone post a proof (if there is)?
Thanks.
Let $f$ and $g$ be Riemann integrable (real) functions and
$f(x)\leq h(x)\leq g(x).$
Is it true that $h(x)$ is Riemann integrable? Can someone post a proof (if there is)?
Thanks.
No.
As an easy counterexample, take any bounded function $h:[0,1] \to \mathbb R$ that is not Riemann integrable. Assuming $a \leqslant h(x) \leqslant b$ for all $x \in [0,1]$, $h$ is always between the constant functions $a$ and $b$, both of which are integrable.
If you want a specific counterexample, take the Dirichlet function $ h(x) = \begin{cases} 1, &x \text{ is rational}, \\ 0, &\text{otherwise}, \end{cases} $ restricted to the unit interval. Clearly, $h$ is bounded between $0$ and $1$.