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Partition of a Closed Interval
Suppose that a,b∈ℝ and a<b. A partition of [a,b] is a finite tuple P=(x0,x1,…,xn) such that a=x0<x1<⋯<xn=b.
Infimum of a Function on a Set
Suppose that f:X→ℝ and A⊆X. Define the infimum of f on A by infA⁡f:=inf⁡{f(x):x∈A}
Supremum of a Function on a Set
Suppose that f:X→ℝ and A⊆X. Define the supremum of f on A by supA⁡f:=sup⁡{f(x):x∈A}.
Lower Riemann Sum
Suppose that f:[a,b]→ℝ is a bounded function, and let P=(x0,x1,…,xn) be a partition of [a,b]. The lower Riemann sum of f over P is L(f,P,[a,b]):=∑j=1n(xj−xj−1)inf[xj−1,xj]⁡f, where inf[xj−1,xj]⁡f is the infimum of f on [xj−1,xj].
Upper Riemann Sum
Suppose that f:[a,b]→ℝ is a bounded function, and let P=(x0,x1,…,xn) be a partition of [a,b]. The upper Riemann sum of f over P is U(f,P,[a,b]):=∑j=1n(xj−xj−1)sup[xj−1,xj]⁡f, where sup[xj−1,xj]⁡f is the supremum of f on [xj−1,xj].
Refinement of a Riemann Partition
Suppose that P and P′ are partitions of [a,b]. We say that P′ is a refinement of P if every point of P is also a point of P′.
Riemann Sum Inequalities under Refinement
Suppose that f:[a,b]→ℝ is bounded, and suppose that P′ is a refinement of P. Then L(f,P,[a,b])≤L(f,P′,[a,b])≤U(f,P′,[a,b])≤U(f,P,[a,b]).
Any Lower Riemann Sum is Below Any Upper Riemann Sum
Suppose that f:[a,b]→ℝ is bounded, and let P and Q be partitions of [a,b]. Then L(f,P,[a,b])≤U(f,Q,[a,b]).
Lower Riemann Integral
Suppose that f:[a,b]→ℝ is bounded. The lower Riemann integral of f over [a,b] is L(f,[a,b]):=supP⁡L(f,P,[a,b]) where P ranges over all partitions of [a,b].
Upper Riemann Integral
Suppose that f:[a,b]→ℝ is bounded. The upper Riemann integral of f over [a,b] is U(f,[a,b]):=infP⁡U(f,P,[a,b]), where P ranges over all partitions of [a,b].
Lower Riemann Integral is Below Upper Riemann Integral
Suppose that f:[a,b]→ℝ is bounded. Then L(f,[a,b])≤U(f,[a,b]).
Riemann Integrable Function
Suppose that f:[a,b]→ℝ is bounded. We say that f is Riemann integrable on [a,b] if L(f,[a,b])=U(f,[a,b]).
Riemann Integral
Suppose that f:[a,b]→ℝ is Riemann integrable. Its Riemann integral over [a,b] is the common value ∫abf:=L(f,[a,b])=U(f,[a,b]).
Continuous Functions on Closed Bounded Intervals are Riemann Integrable
Suppose that a,b∈ℝ and a<b. If f:[a,b]→ℝ is continuous, then f is Riemann integrable on [a,b].
Bounds on a Riemann Integral
Suppose that f:[a,b]→ℝ is Riemann integrable. Then (b−a)inf[a,b]⁡f≤∫abf≤(b−a)sup[a,b]⁡f.
Interchanging Riemann Integral and Pointwise Limit
Suppose that a,b,M∈ℝ, a<b, and f1,f2,… is a sequence of Riemann integrable functions fk:[a,b]→ℝ. Suppose that |fk(x)|≤M for every k∈ℤ+ and every x∈[a,b], and suppose that f(x):=limk→∞⁡fk(x) exists for every x∈[a,b]. If f is Riemann integrable on [a,b], then ∫abf=limk→∞⁡∫abfk.