Post on 17-Dec-2015
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