Chapter 1 - Limits and Continuity

# Chapter 1 - Limits and Continuity - say that f is...

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Limits and Continuity Pg 14 Punctured Neighborhood of a – where we let f be a real-valued function defined (at least) for all points in some open interval containing the point a ϵ R except, possibly, at a itself. Pg 14 (Limit) Given a number L ϵ R, we write to mean: for every ϵ > 0, there is some δ > 0 such that |f(x) – L| < ϵ whenever x satisfies 0 < Ix - al < δ . Pg 15 Theorem 1.17 - Let f be a real-valued function defined in some punctured neighborhood of a ϵ R Then, the following are equivalent: (i) There exists a number L such that (by the ϵ - δ definition). (ii) There exists a number L such that whenever , where for all n. (iii) converges (to something) whenever , where for all n. Pg 15 (Continuous) Now suppose that f is defined in a neighborhood of a, this time including the point a itself. We
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Unformatted text preview: say that f is continuous at a if . That is, if: for every ϵ > 0, there is a δ > 0 (that depends on f, a, and ϵ ) such that |f(x) - f(a)| < ϵ whenever x satisfies |x – a| < δ . Pg 15 Theorem 1.18 - Let f be a real-valued function defined in some neighborhood of a ϵ R. Then, the following are equivalent: (i) f is continuous at a (by the ϵ-δ definition); (ii) whenever ; (iii) converges (to something) whenever . (iv) f(a-) and f(a+) both exist, and both are equal to f(a). Pg 15 Right-continuous at a – if f(a+) exists and equals f(a); Jump Discontinuity at a – if f(a-) and f(a+) both exist but at least one is different from f(a)....
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## This note was uploaded on 11/10/2011 for the course MATH 511 taught by Professor Higdon during the Spring '11 term at Oregon State.

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