Graphics Reference
In-Depth Information
resemblance between parts of its graph and the marks on a
pre-decimal ruler. This 'ruler diagram' illustrates the graph for
rational points where
q
is a power of 2.
(a) By constructing a sequence of irrational numbers tending to
each rational point, prove that
f
is not continuous at any
rational point.
No surprises so far. The really astonishing property of this function
is that it is
continuous
at every irrational point.
(b) Now let
a
bean irrational numbr with 0
a
1.
Given
0, we seek a
such that
x
a
f
(
x
)
f
(
a
)
.
(i) Is there a positive integer
m
such that 1/
m
?
(ii) How many rational numbers
p
/
q
can there be
between 0 and 1 with
q
m
? Might any oneof them
equal
a
?
(iii) Is there a shortest distance
a
p
/
q
for therational
numbers
p
/
q
in part (ii)?
(iv) If wetaketheshortst distancein (iii) as
, how big
?
(v) Does this establish continuity by the neighbourhood
definition at each irrational point?
might
f
(
x
)
f
(
a
)
get when
x
a
One-sided limits
nition of one-sided limits bysequences
73 When examining the discontinuities of the integer function
f
(
x
)
[
x
], we saw that although the sequence (1
1/
n
) tends to 1,
the sequence (
f
(1
1/
n
)) tends to 0 which is not equal to
f
(1).
Construct another sequence (
a
De
fi
) which tends to 1, but for which the
sequence (
f
(
a
)) tends to 0. What property of the sequence (
a
)is
necessary for this to be the case?
