FIRST YEAR CALCULUS
W W L CHEN
c
W W L Chen, 1982, 2005.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
However, this document may not be kept on any information storage and retrieval system without permission
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Chapter 4
CONTINUITY
4.1. Introduction
Example 4.1.1.
Consider the function
f
(
x
) =
x
2
. The graph below represents this function.
It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
Example 4.1.2.
Consider the function
f
(
x
) =
x/

x

, as discussed in Example 3.1.6. If we now attempt
to draw the graph representing this function, then it is impossible to draw this graph without lifting our
pencil from the paper. After all, there is a break, or discontinuity, at
x
= 0, where the function is not
defined. Even if we were to give some value to the function at
x
= 0, then it would still be impossible to
draw this graph without lifting our pencil from the paper. It is impossible to avoid the jump from the
value
−
1 to the value 1 when we go past
x
= 0 from left to right.
Chapter 4 : Continuity
page 1 of 9
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First Year Calculus
c
W W L Chen, 1982, 2005
Example 4.1.3.
Consider the function
f
(
x
) =
x
3
+
x
. We showed in Example 3.1.9 that
f
(
x
)
→
f
(1)
as
x
→
1. The graph represents this function.
As we approach
x
= 1 from either side, the curve goes without break towards
f
(1). In this instance, we
say that
f
(
x
) is continuous at
x
= 1.
We observe from Example 4.1.3 that it is possible to formulate continuity of a function
f
(
x
) at a point
x
=
a
in terms of
f
(
a
) and the limit of
f
(
x
) at
x
=
a
as follows.
Definition.
We say that a function
f
(
x
) is continuous at
x
=
a
if
f
(
x
)
→
f
(
a
) as
x
→
a
; in other
words, if
lim
x
→
a
f
(
x
) =
f
(
a
)
.
Example 4.1.4.
The function
f
(
x
) =
x
2
is continuous at
x
=
a
for every
a
∈
R
.
Example 4.1.5.
The function
f
(
x
) =
x/

x

is continuous at
x
=
a
for every nonzero
a
∈
R
. To see
this, note that for every nonzero
a
∈
R
, there is an open interval
a
1
< x < a
2
which contains
x
=
a
but
not
x
= 0. The function is clearly constant in this open interval.
Example 4.1.6.
The function
f
(
x
) =
x
3
+
x
is continuous at
x
=
a
for every
a
∈
R
.
Example 4.1.7.
The function
f
(
x
) = sin
x
is continuous at
x
=
a
for every
a
∈
R
. To see this, note
first the inequalities

sin
x
−
sin
a

=

2cos
1
2
(
x
+
a
)sin
1
2
(
x
−
a
)
 ≤ 
2sin
1
2
(
x
−
a
)
 ≤ 
x
−
a

(here we are using the well known fact that

sin
y
 ≤ 
y

for every
y
∈
R
). It follows that given any
>
0, we have

f
(
x
)
−
f
(
a
)

<
whenever

x
−
a

<
min
{
, π
}
.
Example 4.1.8.
It is worthwhile to mention that the function
f
(
x
) =
1
if
x
is rational,
0
if
x
is irrational,
is not continuous at
x
=
a
for any
a
∈
R
. In other words,
f
(
x
) is continuous nowhere. The proof is
rather long and complicated. It depends on the well known fact that between any two real numbers,
there are rational and irrational numbers.
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 Spring '09
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 Math, Topology, Continuity, Intermediate Value Theorem, Continuous function, Metric space, W W L Chen

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