Difference between revisions of "The Additivity Theorem"

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Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case, we have that
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'''The Additivity Theorem for Riemann Integrable Functions''': Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case, we have that
  
 
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Revision as of 15:45, 9 November 2021

The Additivity Theorem for Riemann Integrable Functions: Let be a real-valued function on the interval , and let . Then, is Riemann integrable on if and only if it is also Riemann integrable on and . In this case, we have that