Question

Whether the composition of 2 R-S integrable function is R-S in integrable? Justify the answer

Answer 1

Step-by-step explanation:

A function on a bounded interval is Riemann-integrable iff it is bounded and almost everywhere continuous. So the functions

f(x)={10 for x≠0 for x=0 and g(x)={1/q0 for x=p/q for x∉Q

are Riemann-integrable over any bounded interval, since f is continuous everywhere except at 0, and g is continuous at every irrational x. (In the definition x=p/q is the unique representation of rational x with p and q relatively prime integers and q>0.)

The composition of these functions is

f(g(x))={10 for x∈Q for x∉Q

which is nowhere continuous, so not Riemann-integrable over any interval.