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HomeMathematicsCalculusA bounded function ​( f:to mathbb{R} )​ is Riemann integrable iff for...

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 ​\( U(f, P_\epsilon)-L(f, P_\epsilon)<\epsilon. \)

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.
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