10. Let a < b. Let f be a bounded function on [a, b]. You are going to prove that' f is integrable on [a, b] if and only if Ve > 0, there exists a partition P of [a, b] such that Up(f) – Lp(f)< ɛ. (*) (a) Prove that if ƒ satisfies (*), then f is integrable on [a, b]. (b) Prove that if f is integrable on [a, b) then f satisfies (*). 1This is sometimes called the “e-characterization of integrability". It simplifies proofs about integrability. You will use it in MAT237.

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Q: p) State Cauchy's Schwarz Inequality and prove this for Space of Continuous functions on[-n, n].

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.