Theorem :
A bounded function \( f:[a, b]\to \mathbb{R} \) is Riemann integrable iff for every \( \epsilon>0 \) there exist a partition \( P_\epsilon \) of [a, b] such that \( U(f, P_\epsilon)-L(f, P_\epsilon)<\epsilon. \)
Proof :
Suppose f is Riemann integrable on [a, b].
Let \( \epsilon>0 \) be arbitrary.
\( \int\limits_\underline{a}^bf(x)dx=\int\limits_a^\underline{b}f(x)dx \) ….. (1)
By definition of lower integral,
\( \int\limits_\underline{a}^bf(x)dx=sup\{L(P, f) \), P is partition of \( [a, b]\} \)
By definition of supremum,
\( \exists \) a partition \( P_1 \) of [a, b] such that,
\( \int\limits_\underline{a}^bfdx-\frac{\epsilon}{2}<L(P_1, f)\leq\int\limits_\underline{a}^bfdx \) ….. (2)
By definition of upper integral,
\( \int\limits_a^\underline{b}f(x)dx=inf\{U(P, f) \), P is a partition of \( [a, b]\} \)
By definition of infimum,
\( \exists \) some partition\( P_2 \)of [a, b] such that,
\( \int\limits_a^\underline{b}fdx\leq U(P_2, f)<\int\limits_a^\underline{b}fdx+\frac{\epsilon}{2} \) ….. (3)
Let \( P_\epsilon=P_1\cup P_2 \) be the refinement of \( P_1 \) and \( P_2 \).
\( U(P_\epsilon, f)\leq U(P_2, f) \)
\( L(P_1, f)\leq L(P_\epsilon, f) \)
Consider,
\( U(P_\epsilon, f)-L(P_\epsilon, f) \)
\( \leq\int\limits_a^\underline{b}f(x)dx+\frac{\epsilon}{2}-\int\limits_\underline{a}^bf(x)dx+\frac{\epsilon}{2} \) …. from (1), (2) & (3)
\( =\epsilon \)
Hence, we have partition \( P_\epsilon \) such that,
\( U(P_\epsilon, f)-L(P_\epsilon, f)<\epsilon \)
Conversely,
Let \( \epsilon>0 \) be arbitrary and for this \( \epsilon \),
\( \exists \) a partition \( P_\epsilon \) such that, \( U(P_\epsilon, f)-L(P_\epsilon, f)<\epsilon \).
Now, \( U(P_\epsilon, f)\geq L(P_\epsilon, f) \)
\( \implies 0\leq U(P_\epsilon, f)-L(P_\epsilon, f)<\epsilon \)
Since, this inequality true for every \( \epsilon>0 \),
\( \therefore\int\limits_\underline{a}^{b}fdx=\int\limits_a^\underline{b}fdx \)
Hence, f is Riemann integrable.
Also Read: Every monotonic function on [a, b] is Riemann integrable
