10.2: Continuity

Definition of Continuity A function f is continuous at a point x = c if f (c) exists 3. A function f is continuous on the open interval (a,b) if it is continuous at each point on the interval. If a function is not continuous, it is discontinuous.

A constant function is continuous for all x. For integer n > 0, f (x) = x n is continuous for all x. A polynomial function is continuous for all x. A rational function is continuous for all x, except those values that make the denominator 0. For n an odd positive integer, is continuous wherever f (x) is continuous. For n an even positive integer, is continuous wherever f (x) is continuous and nonnegative.

Continuous at -1?Continuous at -2? Continuous at 2?Continuous at 0? YES NOYES NO Continuous at 2? YES Continuous at -4? YES

Discuss the continuity of each function at the indicated points at x=-1 at x=0 at x=1 at x=3 and at x =0 F is continuous at -1 because the limit is 1 and also f(-1)=1 Not continuous at 0 because g(0) is not defined Not continuous at 1 because h(1) is not defined Not continues at 3 because k(3) is not defined and also the limit does not exist Continuous at 0 because the limit is -1 and k(0)=-1