If $f(x)$ and $g(x)$ are Riemann integrable and $f(x)\leq h(x)\leq g(x)$, must $h(x)$ be Riemann integrable?

Solution 1:

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 $1$.

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