3al21_2016.pdf

# 3al21_2016.pdf - 1/19 1 Lecture 19 Continuous Functions 2...

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1/19 1 Lecture 19 Continuous Functions 2 Lecture 20 Continuous Functions II 3 Lecture 21 Continuous Functions III Instructor: David Earn Mathematics 3A03 Real Analysis I

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Lecture 19 Continuous Functions 2/19 Mathematics and Statistics Z M d ω = Z M ω Mathematics 3A03 Real Analysis I Instructor: David Earn Lecture 19 Continuous Functions Wednesday 26 October 2016 Instructor: David Earn Mathematics 3A03 Real Analysis I
Lecture 19 Continuous Functions 3/19 Limits of functions Instructor: David Earn Mathematics 3A03 Real Analysis I

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Lecture 19 Continuous Functions 4/19 Limits of functions Definition (Limit of a function on an interval ( a , b )) Let a < x 0 < b and f : ( a , b ) R . Then f is said to approach the limit L as x approaches x 0 , often written “ f ( x ) L as x x 0 ” or lim x x 0 f ( x ) = L , iff for all ε > 0 there exists δ > 0 such that if 0 < | x - x 0 | < δ then | f ( x ) - L | < ε . Shorthand version: ε > 0 δ > 0 )– 0 < | x - x 0 | < δ = ⇒ | f ( x ) - L | < ε . Instructor: David Earn Mathematics 3A03 Real Analysis I
Lecture 19 Continuous Functions 5/19 Limits of functions The function f need not be defined on an entire interval. It is enough for f to be defined on a set with at least one accumulation point. Definition (Limit of a function with domain E R ) Let E R and f : E R . Suppose x 0 is a point of accumulation of E . Then f is said to approach the limit L as x approaches x 0 , i.e., lim x x 0 f ( x ) = L , iff for all ε > 0 there exists δ > 0 such that if x E , x 6 = x 0 , and | x - x 0 | < δ then | f ( x ) - L | < ε . Shorthand version: ε > 0 δ > 0 )– ( x E 0 < | x - x 0 | < δ ) = | f ( x ) - L | < ε . Instructor: David Earn Mathematics 3A03 Real Analysis I

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Lecture 19 Continuous Functions 6/19 Limits of functions Example Prove directly from the definition of a limit that lim x 3 (2 x + 1) = 7 .
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