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Math 151 Section 2.5 Continuity Continuous A function f ( x ) is said to be continuous at x = a if lim f ( x) = f ( a) . x!a We say f ( x ) is continuous from the left if lim" f ( x) = f ( a) . We say f ( x ) is continuous from the right if lim+ f ( x) = f ( a) . x!a x!a Example: Use the graph of f ( x ) to locate all discontinuities of f ( x ) . For each, identify the limit from the left and the limit from the right and state whether it is continuous from the right, left, or neither. Math 151 Example: Explain why the following functions are not continuous at the indicated value of x. A. f ( x) = !1 (1! x) 2 ,x =1 #% x 2 ! 2x !8 % B. f ( x) = % $ x!4 %% %&3 if x " 4 if x = 4 #2x +1 % % % %3x Example: Identify all points of discontinuity for f ( x) = %$ % 4 % % % 2 % &x + 2 Example: For what value(s) of c is g ( x) continuous, if any? # %x 2 ! c2 g ( x) = % $ % % &cx + 20 if x < 4 if x " 4 if x ! "1 if "1< x <1 if x = 1 if x >1 Math 151 Intermediate Value Theorem If f ( x ) is continuous on the interval [a, b] and N is any number strictly between f ( a) and f (b) , then there exists a number c in (a, b) such that f (c) = N. Example: If g ( x) = x 5 ! 2x 3 + x 2 + 2 , show that there is a number c such that g (c) = !1 . Example: Show that h( x) = x 5 ! 2x 4 ! x ! 3 has a root on the interval (2, 3).