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Unformatted text preview: 5. Continuity P. K. Lamm 09/10/11 (15:22) p. 1 / 8 Lecture Notes: 5. Continuity These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Continuity of Functions Consider the following two limit problems: lim x 3 9- x 2 3- x = lim x 3 (3- x )(3 + x ) 3- x = lim x 3 3 + x = 6 , and lim x 3 x 2 = 3 2 = 9 . In the first example where f ( x ) = 9- x 2 3- x , we have lim x 3 f ( x ) 6 = f (3) because there is no value on the curve at x = 3; thus for the first example there is a hole in the curve above the point x = 3. In the second example where f ( x ) = x 2 , we have lim x 3 f ( x ) = f (3) , so that there is no gap or jump in the curve at x = 3. We say that the function f ( x ) = x 2 is continuous at x = 3, while the curve y = f ( x ) = 9- x 2 3- x is discontinuous or not continuous at x = 3. 1 5. Continuity P. K. Lamm 09/10/11 (15:22) p. 2 / 8 Definition: Suppose f is defined on an interval surrounding the point x . We say f is continuous at x provided (1) f ( x ) exists; (2) lim x x f ( x ) exists; and (3) f ( x ) = lim x x f ( x ) . Definition: We say f is discontinuous at x if f is not continuous at x . Definition: We say f is continuous on ( a,b ) if f is continuous at every point x ( a,b ). Definition: We say f is continuous on [ a,b ] if f is continuous on ( a,b ) and lim x a + f ( x ) = f ( a ) , and lim x b- f ( x ) = f ( b ) . Definition: We say f is continuous if it is continuous on its domain. If a function f is continuous on [ a,b ] then we can use a pencil to draw its graph from ( a,f ( a )) to ( b,f ( b )) without ever having to cause the pencil to leave the page. A continuous function has no holes, gaps, or jumps. Some common continuous functions: 1. Polynomials are continuous on (- , ). 2. The trig functions sin x and cos x are continuous on (- , ). 3. The absolute value function f ( x ) = | x | is continuous on (- , )....
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