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
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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.