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Unformatted text preview: 109 Spring 2011  Supplement on continuity of realvalued 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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 Spring '06
 Knutson
 Math, Continuity, Polynomials, Negative and nonnegative numbers, δ, Xn, R+

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