10.3: Continuity

Definition of Continuity A function f is continuous at a point x = c if 1. 2. 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) = xn 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? YES NO Continuous at 2? YES Continuous at 0? Continuous at 2? YES NO Continuous at -4? YES

Discuss the continuity of each function (-∞,∞) This function is continuous on (-∞,-1)U(-1,3)U(3, ∞) because x can’t equal to -1 or 3 This function is continuous on (-20,∞) because X + 20 ≥ 0 so x ≥ - 20 This function is continuous on (-∞,∞) because the radicand is always ≥ 0 This function is continuous on This function is continuous on (-∞,∞) (-∞,1)U(1, ∞) because x can’t equal to 1 This function is continuous on