Calculus 1 Lecture Notes
Section 1.4
Page 1 of 6
Section 1.4: Continuity and Its Consequences
Big Idea: For many of the functions we looked at yesterday, it seemed that lim f(x)  f(a) for many
x a
values of a. The name for when a function behaves like this at a given value is continuity. Continuity
is related to the appearance of a graph in that a function is continuous on an interval if its graph can be
drawn without lifting your pencil from the paper.
Big Skill: You should be able to identify intervals of continuity, remove discontinuities if possible, and
use the method of bisections to find zeros of a function.
Some examples of functions with discontinuities:
x2  4
is not continuous at
x2
x = 2 (the graph has a hole).
Graph of y  g ( x) 
Graph of y  f ( x) 
x2  5
 x  2
2
is not continuous
at x = 2 (the graph “blows up”).
Calculus 1 Lecture Notes
Section 1.4
x2
is not continuous at
x2
x = 2 (the graph has a jump discontinuity).
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Graph of y  f ( x) 
 sin  x 
for x  0

Graph of y  g ( x)   x
is not
 2
for x  0

continuous at x = 0 (the graph has a hole).
Definition 4.1: Formal definition of continuity at a point
A function f is continuous at x = a when:
i. f(a) is defined.
lim f(x) exists
ii.
x a
iii.
lim f(x)  f(a) (Note this implies i. and ii.)
x a
Otherwise, f is said to be discontinuous at x = a.
(Notice that the third part of the definition implies the first two parts.)
Practice:
1. Determine where g is continuous for g ( x) 
x2  4
.
x2
Some discontinuities can be removed by redefining the function in a piece-wise manner at the point
where the discontinuity occurs.
Practice:
x2  4
2. Make the function g ( x) 
continuous everywhere by redefining it at a single point.
x2
Calculus 1 Lecture Notes
Section 1.4
 sin  x 

3. Remove the discontinuity in the function g ( x)   x
 2

point.
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for x  0
by redefining it at a single
for x  0
x
, and remove as many of the discontinuities
sin  x 
as possible by redefining the function at a specific point or points.
4. Find all discontinuities of the function f  x  
Theorem 4.1 (Some Common Functions and Their Intervals of Continuity):
Polynomial functions and the functions sin(x), cos(x), tan-1(x) and ex are continuous everywhere.
n
x is continuous everywhere when n is an odd integer.
n
x is continuous for x > 0 when n is an even integer.
ln (x) is continuous for x > 0.
sin-1(x) and cos-1(x) are continuous for -1 < x < 1.
Three theorems about the continuity of combinations of functions:
Theorem 4.2 (Continuity of Arithmetic Combinations of Functions):
If f and g are continuous at x = a, then
i. (f  g) is continuous at x = a.
ii. (f  g) is continuous at x = a.
iii. (f / g) is continuous at x = a if g(a)  0 and discontinuous at x = a if g(a) = 0.
Calculus 1 Lecture Notes
Section 1.4
Page 4 of 6


Theorem 4.3 (Limit of a Composition of Functions):
If lim g(x)  L and f is continuous at x = L, then lim f g ( x)   f lim g ( x)  f ( L) .
xa
xa
xa
(If the outer function in a composition of functions is continuous, then the limit can be “brought
inside” the outer function.)
Corollary 4.1 (Continuity of a Composition of Functions):
If g is continuous at x = a and f is continuous at g(a), then (f  g) is continuous at a.
Practice:
x2  5
5. Determine where f is continuous for f ( x)  2
.
x  x2
6. Determine where f is continuous for f ( x) 
x 4  3x 2  2
.
x 2  3x  4
Calculus 1 Lecture Notes
Section 1.4
Page 5 of 6
Definition 4.2 (Formal definition of continuity on different types of intervals):
i. A function f is continuous on the open interval (a, b) if it is continuous at every point
x  (a, b).
ii. A function f is continuous on the closed interval [a, b] if it is continuous on the open interval
(a, b) and lim f  x   f a  and lim f x   f b  .
xa
x b
iii. A function f is continuous everywhere (or simply continuous) if it is continuous at every
point x  (-∞, ∞).
Practice:
7. Determine where f is continuous for f ( x)  x 2  4 .
8. Determine where f is continuous for f ( x)   cos  x  
0.25
.
y





x





ae x  1
for x  0


 x
9. Compute a and b to make f ( x)  sin 1   for 0  x  2 continuous everywhere.
 2

2
 x  x  b for x  2




Calculus 1 Lecture Notes
Section 1.4
Page 6 of 6
Two Consequences/Applications of Continuity
Theorem 4.4 (Intermediate Value Theorem):
If f is continuous on the closed interval [a, b] and W is any number between f(a) and f(b), then there is
at least one number c  [a, b] for which f(c) = W.
Picture:
Corollary 4.2 (A Function has a Zero in an Interval Where the Values of Its Endpoints Change
Signs):
If f is continuous on the closed interval [a, b] and f(a) and f(b) have opposite signs (i.e., f(a) f(b) < 0),
then there is at least one number c  (a, b) for which f(c) = 0 (i.e., c is a zero of f ).
Corollary 4.2 can be used to find numerical approximations to the zeros of a function in a technique
called the method of bisections. The method of bisections involves finding finer and finer subdivisions
of an interval containing a zero such that the value of the function always has opposite signs at each
endpoint of the subdivided interval.
Practice:
10. Use the method of bisections to find the zeros of f ( x)  x 2  x  1 .
11. Use the method of bisections to find the zeros of f ( x)  x 3  2 x 
1
.
2