Theorem: Every Monotone Function on [a,b] is Riemann integrable.
Proof;
A Function is said to be monotonic either it is increasing or decreasing.
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| Every Monotone Function is Riemann integrable |
A Function is said to be increasing if for all y,z ε [a,b] we have y ≤ z then f(y) ≤ f(z).
A Function is said to be monotonic decreasing function if for all y,z ε [a,b] we have y ≤ z then f(y) ≥ f(z).
Related theorems;
Every continuous function on [a,b] is Riemann integrable
How do you prove that Dirichlet function is Riemann Integrable
Prove that lower Riemann integral is less than or equal to upper Riemann integral
Since [a,b] is closed interval so function is bounded and we know every bounded function has maximum and minimum.
To prove every monotone function is Riemann integrable we have to show that a bounded increasing function or a bounded decreasing function is Riemann integrable and this can be done by Cauchy Criterion for Riemann integral.
There are two cases;
Case – 1 If function is bounded increasing;
Let P={a=X•,X1,X2 ,...,X_n=b] be Partition of closed interval [a,b].
Then by definition of bounded increasing function
f(a) ≤ f(X) ≤ f(b) where X ε P
m_r = inf { f(X) : X ε [a,b] } = f(a)
M_r = sup { f(X) : X ε [a,b] } = f(b)
This implies that
U(P , f) = ∑ M_r.δ_r
L(P , f) = ∑ m_r.δ_r
Where r varies from r=0 to r=n.
U(P, f) – L(P, f) = ∑ (M_r - m_r).δ_r
= [f(b) – f(a)] ∑ δ_r
This implies that
U(P, f) – L(P, f) = [ f(b) – f(a) ] (b – a)
Let (b – a) < ε / f(b) – f(a) , this implies that
U(P, f) – L(P, f) < ε
Case – 2 If function is bounded decreasing;
Then by definition of bounded decreasing function
f(a) ≥ f(X) > f(b)
Where X ε Partition P.
m_r = f(b) & M_r = f(a)
This implies that
U(P, f) – L(P, f) = ∑ (M_r - m_r).δ_r
= [ f(a) – f(b) ] ∑ δ_r
This implies that
U(P, f) – L(P, f) = [ f(a) – f(a) ] (b – a)
Let (b – a) < ε / f(a) – f(b) , this implies that
U(P, f) – L(P, f) < ε
Since difference between Upper Riemann Integral and lower Riemann integral is less than epsilon (ε > 0) for both cases, hence proved that every monotone function is Riemann integrable.