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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 R + such that for all x R , < | x x | < = | 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 such that for each x with 0 < | x | < , 0 < | x 2 | < . Define = . Then suppose 0 < | x | < . | x 2 | = x 2 < 2 = ( ) 2 = . Example. The modulus function has a limit 0 at x = 0. Proof. Let R + . We want to find such that for each x with 0 < | x | < , 0 < || x | | < . Define = . Then suppose 0 < | x | < . || x | | = | x | < = . 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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