exercise:D3c4ffc03c: Difference between revisions
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Prove that if <math>f</math> is bounded on <math>(a,b]</math> and integrable | Prove that if <math>f</math> is bounded on <math>(a,b]</math> and integrable over <math>[t,b]</math> for every <math>t</math> in <math>(a,b]</math>, then <math>f</math> is integrable over <math>[a,b]</math> and <math>\lim_{t\goesto a+} \int_t^b f = \int_a^b f</math>. | ||
over <math>[t,b]</math> for every <math>t</math> in <math>(a,b]</math>, then <math>f</math> | |||
is integrable over <math>[a,b]</math> and | [''Hint:'' The argument is essentially the same as that in the proof of [[guide:F22a4e9424#theorem-1|Theorem]].] | ||
<math>\lim_{t\goesto a+} \int_t^b f = \int_a^b f</math>. | |||
[''Hint:'' The argument is essentially the same | |||
as that in the proof of Theorem | |||
Latest revision as of 02:47, 25 November 2024
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[/math]
Prove that if [math]f[/math] is bounded on [math](a,b][/math] and integrable over [math][t,b][/math] for every [math]t[/math] in [math](a,b][/math], then [math]f[/math] is integrable over [math][a,b][/math] and [math]\lim_{t\goesto a+} \int_t^b f = \int_a^b f[/math].
[Hint: The argument is essentially the same as that in the proof of Theorem.]