Continuity
TS: Making decisions after
reflection and review
Objectives
To find the intervals on which a function is
continuous.
 To find any discontinuities of a function.
 To determine whether discontinuities are
removable or non-removable.

Video Clip from
Calculus-Help.com
Continuity
What makes a function continuous?

Continuous functions are predictable…
1) No breaks in the graph
A limit must exist at every x-value or the
graph will break.
2) No holes or jumps
The function cannot have undefined
points or vertical asymptotes.
Continuity

Key Point:
Continuous functions
can be drawn with a
single, unbroken
pencil stroke.
Continuity

Mathematically speaking…
If f (x) is continuous, then for every x = c
in the function, lim f ( x)  f (c)
x c

In other words, if you can evaluate any
limit on the function using only the
substitution method, then the function is
continuous.
Continuity of Polynomial and
Rational Functions

A polynomial function is continuous at
every real number.

A rational function is continuous at every
real number in its domain.
Polynomial Functions
f ( x)  x  x
3

f ( x)  x  2 x  3
2
Both functions are continuous on ( , ) .
Rational Functions
1
f ( x) 
x
x2  1
f ( x) 
x 1
continuous on:
continuous on:
(, 0)  (0, )
(, 1)  (1, )
Rational Functions
1
f ( x)  2
x 1
1
f ( x)  2
x 1
continuous on:
continuous on:
(, )
(,  1)  (1, 1)  (1, )
Piecewise Functions
 x 2  4, x  2
f ( x)  
 x  2, x  2
22  4  4  4  0

2  2  0
continuous on
(, )
Discontinuity

Discontinuity: a point
at which a function is
not continuous
Discontinuity

Two Types of Discontinuities
1) Removable (hole in the graph)
2) Non-removable (break or vertical
asymptote)

A discontinuity is called removable if a
function can be made continuous by
defining (or redefining) a point.
Two Types of Discontinuities
Discontinuity

Find the intervals on which these function are
continuous.
x2
f ( x)  2
x  3x  10
x2

( x  2)( x  5)
1

( x  5)
Point of discontinuity:
x20
x  2
Removable
discontinuity
Vertical Asymptote:
x 5  0
x5
Non-removable
discontinuity
Discontinuity
x2
f ( x)  2
x  3x  10
Continuous on:
(,  2)  (2, 5)  (5, )
Discontinuity
2 x, x  2
f ( x)   2
 x  4 x  1, x  2
lim(
2 x)  4

x2
lim ( x 2  4 x  1) 3
x 2
f (2)  4
Continuous on:
(, 2]  (2, )
Discontinuity

Determine the value(s) of x at which the
function is discontinuous. Describe the
discontinuity as removable or non-removable.
(A)
x2  1
f ( x)  2
x  5x  6
(C)
x  4x  5
f ( x) 
2
x  25
x  10 x  9
f ( x) 
x 2  81
2
(B)
2
x 4
f ( x)  2
x  2x  8
2
(D)
Discontinuity
x 1
f ( x)  2
x  5x  6
2
(A)
( x  1)( x  1)

( x  6)( x  1)
x  1
x6
Removable discontinuity
Non-removable discontinuity
Discontinuity
x  10 x  9
f ( x) 
2
x  81
2
(B)
( x  9)( x  1)

( x  9)( x  9)
x  9
x9
Removable discontinuity
Non-removable discontinuity
Discontinuity
x  4x  5
f ( x) 
2
x  25
2
(C)
( x  5)( x  1)

( x  5)( x  5)
x5
x  5
Removable discontinuity
Non-removable discontinuity
Discontinuity
(D)
x2  4
f ( x)  2
x  2x  8
( x  2)( x  2)

( x  4)( x  2)
x  2
x4
Removable discontinuity
Non-removable discontinuity
Conclusion

Continuous functions have no breaks, no
holes, and no jumps.

If you can evaluate any limit on the
function using only the substitution
method, then the function is continuous.
Conclusion
A discontinuity is a point at which a
function is not continuous.
 Two types of discontinuities
 Removable (hole in the graph)
 Non-removable (break or vertical
asymptote)
