How to prove a function is Riemann integrable

To prove a function is Riemann integrable we have to show that

Lower Riemann integral = Upper Riemann Integral

                                  Or

U(P , f) – L(P , f) < ε  ( Cauchy Criterion)

These are two methods/conditions to prove a function is Riemann integrable and choice of condition to prove a function is Riemann integrable or not depends on function because function may be constant or not. 

If function is constant then to prove that function is Riemann integrable we can choose first method to prove that function is Riemann integrable or not for which we have to show that “ Lower Riemann integral is equal to Upper Riemann integral “ and the reason behind choice of this method to prove a constant function is Riemann integrable or not is that the Supremum and infimum of constant function is function itself and if constant function is not piecewise function then it’s supremum and infimum are equal as a result lower Riemann sum and upper Riemann sum is equal this implies that supremum of Lower Riemann sum is equal to infimum of Upper Riemann sum. Since

Lower Riemann integral = sup { L(P,f) : P€ S }

Upper Riemann integral = inf { U(P,f) : P€ S }

Where S denotes the set of partitions.

And as a result a function that is not piecewise function is Riemann integrable.

But if constant function is piecewise function then function may be or not Riemann integral and it depends on the condition that lower Riemann integral is equal to upper Riemann integral or not if yes then piecewise constant function is Riemann integrable if no then it is not Riemann integrable.

If function is not a constant function then we can choose second condition that is

U(P , f) – L(P ,f ) < ε

to prove a function that is not constant function is Riemann integrable or not and this condition is called Cauchy Criterion to check whether a function is Riemann integrable or not.