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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
(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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern