10.2: 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.

)()(lim cfxfcx

exists)(lim xfcx

• 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.

n xf )(

n xf )(

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

3

3)(

1

1)(

2)(

32)(

2

x

xxk

x

xxh

xxg

xxf 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