Theorem: If f is bounded function and has only a finite number of points of discontinuity on [a,b] then function is Riemann integrable.
Proof;
Since given that function is bounded and has finite number of points of discontinuity on [a,b], let suppose there are four numbers of points of discontinuity on [a,b].
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| If f is bounded function and has only a finite number of points of discontinuity on [a,b] then function is Riemann integrable |
Firstly considering those intervals which contains points of discontinuity a1, a2, a3 and a4,
I’1 = [a’1, a”1] ⇒ δ’1 = a”1 – a’1
I’2 = [a’2, a”2] ⇒ δ’2 = a”2 – a’2
I’3 = [a’3, a”3] ⇒ δ’3 = a”3 – a’3
I’4 = [a’4, a”4] ⇒ δ’4 = a”4 - a'4
This implies that
Area of upper Rectangles =∑ M’_r.δ’_r
Area of Lower Rectangles =∑ m’_r.δ’_r
Area of upper Rectangles – Area of Lower Rectangles =∑ (M’_r – m’_r)δ’_r
Since
M'_r - m'_r < M_r - m_r
This implies that
Area of upper Rectangles – Area of Lower Rectangles < ∑ (M_r - m_r)δ’_r
Let suppose ∑ δ’_r = ε’
This implies that
Area of upper Rectangles – Area of Lower Rectangles < (M_r - m_r) ε’
Secondly considering those intervals on which function is continuous,
I1”1 = [a, a’1] ⇒ δ”1 = a’1 – a
I”2 = [a”1, a’2] ⇒ δ”2 = a’2 – a”1
I”3 = [a”2, a’3] ⇒ δ”3 = a’3 – a”2
I”4 = [a”3, a’4] ⇒ δ”4 = a’4 – a”3
I”5 = [a”4, b] ⇒ δ”5 = b – a”4
This implies that
Area of upper Rectangles = ∑ M”_r.δ”_r
Area of Lower Rectangles = ∑ m”_r.δ”_r
Area of upper Rectangles – Area of Lower Rectangles = ∑ (M”_r – m”_r)δ”_r
Since M”_r – m”_r < M_r - m_r
Assume that ∑ δ_r = ε”
This implies that
Area of upper Rectangles – Area of Lower Rectangles < (M_r - m_r) ε”
Lastly, after summing up areas of all upper Rectangles and lower Rectangles we have
Area of all upper Rectangles – Area of all Lower Rectangles = U(P, f) – L(P, f) < (M_r - m_r)ε’ + (M_r - m_r)ε”
Suppose ε’ = ε / 2(M_r - m_r) and ε” = = ε / 2(M_r - m_r)
This implies that