The Additivity Theorem

The Additivity Theorem for Riemann Integrable Functions: Let $f$ be a real-valued function on the interval $[a,b]$, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\in (a,b) } . Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } is Riemann integrable on $[a,b]$ if and only if it is also Riemann integrable on $[a,c]$ and $[c,b]$. In this case,

$\int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f$