2.5 CONTINUITY
Intuitively, the graph of a function can be described as a “continuous curve” if
it has not breaks or holes.
The graph of a function has a break or hole if any of the following conditions
occur:
• The function f is undefined at c
•The limit of f(x) does not exist as x approaches c
•The value of the function and the value of the limit at c are different.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Continuous Function
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Example: Determine whether the following functions are continuous at x=-3.
 x2  9
 x2  9
, x  3
, x  3
x 9


f ( x) 
, g ( x)   x  3
, h( x )   x  3
x3
0,
6,
x  3
x  3


2
Solution:
Observe that
x2  9
( x  3)( x  3)
lim f ( x)  lim g ( x)  lim h( x)  lim
 lim
 lim( x  3)  6
x3
x3
x3
x3 x  3
x3
x3
x3
•f(x) is not continuous at x=-3 since it’s undefined at x=-3,
g ( x)  g (3)
•g(x) is not continuous at x=-3 since xlim
3
•h(x) is continuous
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Continuity on an interval
If a function f is continuous at each number in an open interval (a, b),
then we say that f is continuous on (a, b). In the case where f is
continuous on (, ) , we say f is continuous everywhere.
A function is continuous from the left at c if lim f ( x)  c
x c
A function is continuous from the right at c if
lim f ( x)  c
x c 
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Continuous on a closed interval
2.5.2 Definition.
A function f is said to be continuous on a closed interval [a, b] if the following
conditions are satisfied:
1. f is continuous on (a, b)
2. f is continuous from the right at a
3. f is continuous from the left at b.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Some properties of continuous functions
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Continuity of polynomials and rational functions.
2.5.4 Theorem
(a) A polynomial is continuous everywhere.
(b) A rational function is continuous at every point where the denominator is
nonzero, and has discontinuities at the points where the denominator is
zero.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Ex: For what values of x is there a discontinuity in the graph of
x2  4
y 2
x  5x  6
Solution: The function is a rational function, and hence is continuous at
every number where there denominator is nonzero.
Solve the equation
x2  5x  6  0
Yields discontinuities at x=2 and x=3.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
For example. Find a value of the constant k, if possible, that will make the
function continuous everywhere.
7 x  2, x  1
f ( x)   2
kx , x  1
Solution: since 7x-2 and kx2 are both polynomials, f is continuous for x<1
and x>1. x=1 is the only possible discontinuity for f(x).
2
lim f ( x)  lim(7
x

2)

5,
lim
f
(
x
)

lim
kx
 k,



x1
x1
x1
x1
So if k=5, then f is continuous for all x.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Continuity of compositions
For example, lim | g ( x) || lim g ( x) | provided lim g ( x) exists
xc
xc
x c
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
2.5.6 Theorem
(a) If the function g is continuous at c, and the function f is continuous
at g(c), then the composition f g is continuous at c.
(b) If the function g is continuous everywhere and function f is
continuous everywhere, then the composition f g is continuous
everywhere.
The absolute value of a continuous function is continuous.
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
The intermediate-value theorem
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Approximating roots using the intermediate-value theorem
2.5.8 Theorem.
If f is continuous on [a, b], and if f(a) and f(b) are nonzero and have opposite
signs, then there is at least one solution of the equation f(x)=0 in the interval
(a, b).
Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.