Intro+to+the+derivative+notes.pdf

Let us start by noticing that sometimes the limit of

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Let us start by noticing that sometimes the limit of a function y = f ( x ) as x approaches a can be often be found simply by calculating f ( a ). In such a case, the function f is said to be continuous at a . More precisely: Definition 3.14. Let f be a function and a a number in the domain of f . Then we say that f is continuous at a if C1. lim x a f ( x ) exits , and C2. lim x a f ( x ) = f ( a ). The function f is said to be continuous on its domain if it is continuous at each point in its domain. If f is not continuous at a we say that f is discontinuous at a or that f has a discontinuity at a . Thus, a discontinuity can occur at x = a if either D1. lim x a f ( x ) does not exits , or D2. lim x a f ( x ) exists but it is not equal to f ( a ). a. a 0 f ( a ) x y b. a 0 g ( a ) x y c. a 0 x y
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3. Introduction to the derivative: Limits and Continuity 15 The function f shown in Figure a. is continuous at x = a , while functions g and h whose graphs are shown in Figures b. and c. , respectively are not continuous at a . Note that lim x a g ( x ) 6 = g ( a ) and lim x a h ( x ) does not exist. Intuitively, a function f is continuous at a number a in its domain a if we are able to draw the portion of the graph of f near x = a without lifting our pencil from the paper. There are a lot of continuous functions (on their respective domains)! Polynomials, rational functions, exponential functions, logarithms, radicals, absolute values, trigonometric functions, and combinations of those are continuous functions. Example 3.15. Which of the following functions are continuous on their domain? a. f ( x ) = x + 1 if x 1 3 - x if x > 1 b. g ( x ) = 1 /x c. h ( x ) = 1 /x if x 6 = 0 0 if x = 0 Solution: a. The function f is continuous at every point of its domain, because for a < 1, the function f coincides with y = x +1, which is continuous at a . If a > 1, f equals y = 3 - x which is continuous at a . It remains to see what happens when x = 1. Looking at the graph of f , we can conclude that f is continuous at 1, and thus f is continuous on its domain, that is, R . 1 2 0 2 x y
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16 J. S´ anchez-Ortega Proceeding algebraically, from lim x 1 - f ( x ) = lim x 1 ( x + 1) = 1 + 1 = 2 , lim x 1 + f ( x ) = lim x 1 (3 - x ) = 3 - 1 = 2 , we can conclude that lim x 1 f ( x ) exits and equals to f (1) = 2. Therefore, f is continuous at 1. b. The function g ( x ) = 1 /x is continuous on its domain R -{ 0 } because it is a rational function. Notice that 0 is not in the domain of g , so it is not make sense to study the continuity of g at x = 0. c. The domain of h is R . Since h coincides con g except at x = 0, and g is a continuous function on R - { 0 } , we can conclude that h is continuous at a if a 6 = 0. On the other hand, keeping in mind that lim x 0 1 /x does not exist, we have that h is discontinuous at 0. Definition 3.16. Let f be a function and a a number in the domain of f . Suppose that f is discontinuous at x = a . We say that f has a removable discontinuity at x = a if lim x a f ( x ) exists but lim x a f ( x ) 6 = f ( a ). It is called removable because we can remove the discontinuity by redefining f at just the single number a .
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