Difference between revisions of "The Additivity Theorem"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 1: Line 1:
 
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
  
'''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
+
'''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,  
  
<math> </math>
+
<math> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math>
  
 
</blockquote>
 
</blockquote>

Revision as of 15:46, 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,