lecture11 - Lecture 11 We review some basic properties of...

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Lecture 11 We review some basic properties of the Riemann integral. Let Q R n be a rectangle, and let f, g : q R be bounded functions. Assume that f, g are R. integrable. We have the following properties of R. integrals: Linearity: a, b R = af + bg is R. integrable and af + bg = a f + b g. (3.82) Q Q Q Monotonicity: If f g , then g. (3.83) f Q Q Maximality Property: Let h : Q R be a function defined by h ( x ) = max( f ( x ) , g ( x )). Theorem 3.14. The function h is R. integrable and �� h max f, g . (3.84) Q Q Q Proof. We need the following lemma. Lemma 3.15. If f and g are continuous at some point x 0 Q , then h is continuous at x 0 . Proof. We consider the case f ( x 0 ) = g ( x 0 ) = h ( x 0 ) = r . The functions f and g are continuous at x 0 if and only if for every � > 0, there exists a δ > 0 such that f ( x ) f ( x 0 ) < � and g ( x ) g ( x 0 ) < � whenever x x 0 < δ . | | | | | | Substitute in f ( x 0 ) = g ( x 0 ) = r . The value of h ( x ) is either f ( x ) or g ( x ), so h ( x ) r < � for x x 0 < δ . That is h ( x ) h ( x 0 ) < � for x x 0 < δ , so h | | | | | | | | is continuous at x 0 . The proofs of the other cases are left to the student. We defined h = max( f, g ). The lemma tells is that h is integrable.
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