Section1_3

# Section1_3 - Section 1.3 Continuity Overview In Section 1.1...

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(3/19/08) Section 1.3 Continuity Overview: In Section 1.1 we studied one- and two-sided finite limits. In this section we discuss one- and two-sided continuity and continuity on intervals. These concepts will be needed in later chapters for discussions of optimization problems, the definition of definite integrals, and other basic topics. Then we present the Intermediate Value Theorem, which is used in studying equations involving functions that are continuous on finite, closed intervals. Topics: Continuity at a point Continuity of polynomials and rational functions Continuity on intervals Continuity of functions given by single formulas The Intermediate Value Theorem Continuity at a point The function in Figure 1 is continuous from the left at the point a , the function in Figure 2 is continuous from the right at a , and the function of Figure 3 continuous at a , according to the following definition. x y y = f ( x ) a x y y = f ( x ) a x y y = f ( x ) a f is continuous from the left at a. f is continuous from the right at a . f is continuous at a. FIGURE 1 FIGURE 2 FIGURE 3 Definition 1 (One-sided and two-sided continuity at a point) (a) y = f ( x ) is continuous from the left at a if it is defined in an interval ( b, a ] to the left of a , if lim x a - f ( x ) exists, and if lim x a - f ( x ) = f ( a ) . (b) y = f ( x ) is continuous from the right at a if it is defined in an interval [ a, b ) to the right of a , if lim x a + f ( x ) exists, and if lim x a + f ( x ) = f ( a ) . (c) y = f ( x ) is continuous at a if it is defined in an open interval containing a and is continuous from the left and from the right at a . This means that lim x a f ( x ) exists and lim x a f ( x ) = f ( a ) . 20

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Section 1.3, Continuity p. 21 (3/19/08) In each case of Definition 1, three things must happen: the function must be defined at a , the limit must exist, and the limit must equal the value of the function. In the case of continuity from the left in Figure 1, the value of the function at a is given by the dot, and the point on the graph for x < a approaches the dot as x approaches a from the left. In the case of continuity from the right in Figure 2, the value of the function at a is given by the dot, and the point on the graph for x > a approaches the dot as x approaches a from the right. In the case of two-sided continuity in Figure 3, the value of the function at a is given by the dot, and the point on the graph for x negationslash = a approaches the dot as x approaches a from either side. Notice that a function is continuous at a point if it is continuous from the left and from the right at that point. Example 1 Figure 4 shows the graph of a function K , defined by K ( x ) = braceleftbigg x + 4 for - 2 x < 1 x + 1 for 1 x 4 . Use the graph to determine (a) the values a with - 2 a 4 where y = K ( x ) is continuous and (b) whether the function is continuous from the right or from the left at the points in [ - 2 , 4] where it is not continuous.
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