Prove that every continuous function on a closed interval [a,b] is Riemann integrable

Proof;

To prove that every continuous function on a closed interval is Riemann integrable we have to show that

U(P , f) – L(P , f) < ε 

Above inequality is Cauchy Criterion for Riemann integral and it is stated that a function is Riemann integrable on [a,b] if 

U(P,f)-L(P,f) < ε

Every continuous function on a closed interval is Riemann integrable

Since we have given that the function is continuous on (Compact Set) closed interval [a,b] then function is uniformly continuous on closed interval [a,b].

By Definition of uniformly continuous function on closed interval [a ,b] , For each ε > 0 there exists a number δ > 0 such that for all X , Y belong to closed interval [a,b] If 

| X – Y | < δ 

then

| f(X) – f(Y) | < ε

Let P = { a=X.,X1,X2,...,b=Xn} be the partition on closed interval [a,b] then we have 

U(P , f) = ∑ Mr δr

L(P , f) = ∑ mr δr

Where δr = |b - a| and Mr is Supremum of function on closed interval [a,b], mr is infimum of function on closed interval [a,b] and r varies from 1 to n.

Here 

U(P,f)-L(P,f) = ∑(Mr-mr) δr

Where r varies from 1 to n.

Since

           Mr – mr ≥ 0

This implies that 

U(P,f)-L(P,f) = ∑ |Mr-mr|δr

Let [Xr-1 , Xr] belong to closed interval [a,b] and there exists X, Y belong to sub interval [Xr-1 , Xr].

Then 

F(X) = Mr and F(Y) = mr

This implies that

F(X) – F(Y) = Mr – mr

This implies that

U(P,f)-L(P,f) = ∑|F(X)-F(Y)|δr

By Definition of Norm of Partition P , Norm of Partition P is denoted by ||P|| and is defined by

||P|| = max {δr : r=1,2,...,n}

This implies that

{δr : r=1,2,...,n} ≤ ||P||

Since

|X-Y|≤|Xr - Xr-1|≤||P||

Let 

        ||P|| < δ

This implies that

|X-Y| < δ

Then by definition of uniformly continuous function on closed interval [a,b] there exists  ε' > 0 such that

|F(X)-F(Y)| < ε'

This implies that

U(P,f)-L(P,f) < ε’ ∑δr

U(P,f)-L(P,f)< ε’ ∑|b-a|

Where r= 1,2,3,...,n

Suppose that

             ε’< ε/|b-a|

This implies that

U(P , f) – L (P , f) < ε

Hence Proved that every continuous function on closed interval is Riemann integrable.