Graphics Reference
In-Depth Information
thefunction
f
is sequentially continuous at
a
, then
f
is neighbourhood
continuous at
a
.
Neighbourhood continuity is in fact used by many authors as the
basic definition of continuity. In contrast to the sequence definition,
where the infinite process is obvious, the
neighbourhood
definition of continuity looks deceptively finite. The infinite process is
hidden in the fact that the condition must hold for
any
, which must
be thought to take an infinity of values, and in particular to get
arbitrarily small.
There are two simple properties (that will be needed in the next
chapter) which hold for a function continuous at a point, which are
easily established with the neighbourhood definition of continuity.
69 If a function
f
is continuous at a point
a
, by choosing
1, prove
that
f
(
x
) is bounded (above
and
below) in some neighbourhood of
a
.
70 If a function
f
is continuous at a point
a
and
f
(
a
)
0, by a
judicious choiceof
0 in somenighbourhood of
a
. Stateand provean analogous rsult in case
f
(
a
)
0.
, provethat
f
(
x
)
The last question in this section (qn 72) is designed to expose one
of the most counter-intuitive possibilities that follows from the
definition of continuity, namely that the points at which a function is
continuous and thepoints at which a function is not continuous may
be densely packed together in the domain of the function.
As a preparation for a new idea to be used in qn 72 we offer qn 71
as an option.
(71) Find a neighbourhood of
2 which contains no numbers of the
form
n
/2 for any integer
n
.
Find a neighbourhood of
2 which contains no numbers of the
form
n
/3 for any integer
n
.
Find a neighbourhood of
2 which contains no numbers of the
form
n
/4 for any integer
n
.
Find a neighbourhood of
2 which contains no numbers of the
form
n
/5 for any integer
n
.
Find a neighbourhood of
2 which contains no numbers of the
form
n
/2,
n
/3.
n
/4 or
n
/5 for any integer
n
.
72 (
Thomae
, 1875) A function
f
: (0, 1]
R is defined by
0
1/
q
when
x
is irrational,
when
x
is rational and
x
p
/
q
in lowest terms.
f
(
x
)
This function is sometimes called the
ruler function
because of the
