Let f(x) and g(x) be increasing and Riemann integrable. h is defined as h=f-g. Show that h is Riemann integrable. Solution f(x) is increasing and Riemann integrable. g(x) is also increasing and Riemann integrable. As f(x) and g(x) are both Riemann integrable, we can say the limit of the sum of the partitioned area exists. This again implies the difference of the function also is Riemann integrable. As f(x) - g(x) also will have limit as L1-L2 where L1 is Riemann limit of f(x) and L2 is Riemann limit of g(x) Thus h(x) = f(x) - g(x) also will have a limit L1-L2 Hence h(x) is Riemann integrable..

1. Let f(x) and g(x) be increasing and Riemann integrable. h is defined as h=f-g. Show that h is
Riemann integrable.
Solution
f(x) is increasing and Riemann integrable.
g(x) is also increasing and Riemann integrable.
As f(x) and g(x) are both Riemann integrable, we can say the limit of the sum of the partitioned
area exists.
This again implies the difference of the function also is Riemann integrable.
As f(x) - g(x) also will have limit as L1-L2 where L1 is Riemann limit of f(x) and L2 is Riemann
limit of g(x)
Thus h(x) = f(x) - g(x) also will have a limit L1-L2
Hence h(x) is Riemann integrable.