MATH 109

# MATH 109 - 109 Spring 2011 Supplement on continuity of...

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Unformatted text preview: 109 Spring 2011 - Supplement on continuity of real-valued functions. Earlier, we defined functions in general (Eccles chapter 8). An important special case consists of functions on the reals, f : R → R . Example. • Polynomials, exponentials, trigonometric functions • Modulus function | x | = ( x if x ≥ − x if x < • Step function H ( x ) = ( i f x < 1 i f x ≥ • Floor and ceiling functions b x c = largest integer less than x d x e = smallest integer greater than x Definition. Let f : R → R and x ∈ X . Then f has a limit L at x if for each ∈ R + , there is a δ ∈ R + such that for all x ∈ R , < | x − x | < δ = ⇒ | f ( x ) − L | < . Example. The function given by f ( x ) = x 2 has a limit 0 at x = 0. Proof. Let ∈ R + . We want to find δ such that for each x with 0 < | x | < δ , 0 < | x 2 | < . Define δ = √ . Then suppose 0 < | x | < δ . | x 2 | = x 2 < δ 2 = ( √ ) 2 = . Example. The modulus function has a limit 0 at x = 0. Proof. Let ∈ R + . We want to find δ such that for each x with 0 < | x | < δ , 0 < || x |− | < . Define δ = . Then suppose 0 < | x | < δ . || x | − | = | x | < δ = . Example. The function given by f ( x ) = ( | x | x if x = 0 i f x = 0 does not have a limit at 0. Proof. Notice that if x < 0 then f ( x ) = − 1 and if x > 0 then f ( x ) = 1. We will show that for any L ∈ R , L is not the limit of f at x = 0. Let L be any number and let = 1....
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## This note was uploaded on 08/02/2011 for the course MATH 109 taught by Professor Knutson during the Spring '06 term at UCSD.

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MATH 109 - 109 Spring 2011 Supplement on continuity of...

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