Question

Show that 𝑈(−𝑓, 𝑝) = −𝐿(𝑓, 𝑝) and 𝐿(−𝑓, 𝑝) = −𝑈(𝑓, 𝑝). 

Answer 1

𝑈(𝑓, 𝑝) and 𝐿(𝑓, 𝑝) are upper and lower Riemann sums for the partition p.

$𝑈(𝑓, 𝑝)=\sum M_i\Delta x_i,\ L(𝑓, 𝑝)=\sum m_i\Delta x_i$

$M_i=sup\{f(x):x_{i-1}\le x \le x_i\},\ m_i=inf\{f(x):x_{i-1}\le x \le x_i\}$

$sup\{-f(x):x_{i-1}\le x \le x_i\}=-inf\{f(x):x_{i-1}\le x \le x_i\}=-m_i$

$U(-f,p)=-\sum m_i\Delta x_i=-L(f,p)$

$inf\{-f(x):x_{i-1}\le x \le x_i\}=-sup\{f(x):x_{i-1}\le x \le x_i\}=-M_i$

$L(-f,p)=-\sum M_i\Delta x_i=-U(f,p)$