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1 Math 1431 Dr. Melahat Almus Visit CASA regularly for announcements and course material! If you me, please mention your course (1431) in the subject line. Access code deadline: Purchase popper scantrons from UH Bookstore. Did you take Practice test 1? Quiz 1? Quiz 2? Quiz in LAB (on Fridays) EMCF 1 due this week. Respect your friends! Do not distract anyone by chatting with people around you Be considerate of others in class. 1
2 Section 1.4 Continuity Definition: Let c be a real number and f be a function. We say that f is continuous at c if lim f f c. c The function is said to be discontinuous at c if it is not continuous there. According to this definition, the function f is continuous at c if all of the following conditions are met: (i) f is defined at c, (ii) lim f eists, and c lim f f c. c (iii) If any of these conditions fails, the function is not continuous atc. We may say that the function is discontinuous at c. 2
3 When is a function discontinuous at c? 1) If f is not defined at c, we know that graph has a hole or an asymptote at c and the function is not continuous there. 2) If f is defined at c (that is, c is in the domain of f ), then f can be discontinuous for one of these reasons: a) lim f c lim f c does not eist, b) eists, but lim f f c c. Types of discontinuity: If a function f is discontinuous atc, this discontinuity can be classified as: Removable discontinuity if (i) (ii) f is not defined at c, or f c and lim f c eist, but lim f f c c. 3
4 Jump discontinuity if each one-sided limit eists but they are not equal. Infinite discontinuity if f generally associated with having a vertical asymptote at on at least one side of c. This type is c. 4
5 Geometrically speaking a function is continuous if the graph has no holes or breaks. That is, you can trace the graph without removing your pen. Are the following functions continuous? 5
6 Eample: Study the continuity of the function and classify each point of discontinuity as jump, removable, or infinite. 6
7 What if the function is defined by a formula? f 1. Eample: Find the points of discontinuity (if any): 3 2 Eample: Find the points of discontinuity (if any): 2 f 2 4. Eample: Find the points of discontinuity (if any): f sin2. 7
8 Fact: The following types of functions are continuous at every number in their domains: Polynomials, Rational functions, Root functions, Trigonometric functions, Inverse trigonometric functions, Eponential functions, Logarithmic functions. 3 Eample: When is f discontinuous? Eample: When is 2 f discontinuous? Eample: When is f cos discontinuous? Eample: When is f e discontinuous? 8
9 Many complicated continuous functions can be built up using simple ones. Theorem 1.4.1: If f and g are continuous at c, then (i) f g is continuous at c, (ii) f g is continuous at c, (iii) kf is continuous at c (where k is any real number), (iv) fg is continuous at c, (v) f /g is continuous at c, provided gc 0. Parts (i) (iv) can be etended to any finite number of functions. Eample: Find the points of discontinuity (if any): f. 1 2sin 9
10 EXTRA - We studied one-sided limits in Section 1.2; similarly, we may consider onesided continuity. Definition: A function f is said to be continuous from the left at c if lim f f c. c It is continuous from the right at c if lim f f c c. Eample: Is the following function continuous at =0? Is it continuous from the right at =0? Is it continuous from the left at =0? 10
11 Continuity over an interval Definition: Let a,b be an open interval. A function is said to be continuous over a,b if it is continuous at every number in this interval. If f is defined on a closed interval a,b, we only epect to have one-sided continuity at the end points a and b. That is, if the function is continuous at every number in a,b, continuous from the right at a and continuous from the left at b, then we say that the function is continuous over a,b. Eample: Find the interval(s) over which the function f 5 is continuous. Eample: Find the interval(s) over which the function 2 f 1 9 is continuous. Eample: Find the interval(s) over which the function 2 f 2 is continuous. 11
12 How to work with piece-wise functions: Eample: Find all points of discontinuity and classify them: 2 1, if 0 f 1, if 0 2 1, if 2 5 Is this function continuous at =0? (i) Is f (0) defined? (ii) Does lim f ( ) eist? 0 0 lim f( )? lim f( )? 0 (iii) Is f (0) equal to lim f ( )? 0 Is this function continuous at =2? (iv) Is f (2) defined? (v) Does lim f ( ) eist? 2 2 lim f( )? lim f( )? 2 (vi) Is f (2) equal to lim f ( )? 2 12
13 Eercise: Find all points of discontinuity and classify them: f 1, if 1, if 1 4 2, if , if 5 1 Eample: Find the values of A and B so that the function is continuous everywhere. 2 A 14, if 2 f 10, if 2 B, if 2 13
14 Did you take Practice Test 1? Quiz 1? Read Section 1.4 from your tet book. Check CASA regularly for announcements. 14
Section MWF 12 1pm SR 117
Math 1431 Section 1485 MWF 1 1pm SR 117 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus COURSE WEBSITE: http://www.math.uh.edu/~almus/1431_sp16.html Visit my website regularly for announcements
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