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Which of the four sequences
(i) (
a
), (ii) (
b
), (iii) (
f
(
a
)), (iv) (
f
(
b
))
are null sequences?
Deduce that
f
is not continuous at
x
0.
If
f
(0) had some value other than 0, might this redefined function
f
becontinuous at
x
0?
20 (
Dirichlet
, 1829) Define a function
f
: R
R by
f
(
x
)
0 when
x
is
irrational, and
f
(
x
)
1 when
x
is rational.
If
a
is a rational number, show that
f
is not continuous at
x
a
by
considering the sequence (
a
2/
n
.
If
a
is an irrational number, show that
f
is not continuous at
x
a
by considering the sequence ([10
), where
a
a
a
]/10
), theinfinitedcimal
sequence for
a
, of qn 3.51.
Question 18 gives us an example of a function with an infinity of
isolated discontinuities, qn 19 an example of a function with a single
discontinuity, and qn 20 an example of a function with a dense set of
discontinuities. This will be enough to start with, and we now turn to
theformal confirmation of continuity in thefamiliar context of
polynomials.
Sums and products of continuous functions
21 If a real function
f
is defined by
f
(
x
)
c
, that is to say,
f
is a
constant function, provethat
f
is continuous at each point of its
domain.
x
, so that
f
is theidentity
function, provethat
f
is continuous at each point of its domain.
22 If a real function
f
is defined by
f
(
x
)
23
The sum rule
If real functions
f
and
g
havethesamedomain
A
, and areboth
continuous at
x
a
A
, useqn 3.54(iii), thesum rul, to provethat
thereal function
f
g
defined by
(
f
g
)(
x
)
f
(
x
)
g
(
x
)
is also continuous at
x
a
.
24 Useqns 21, 22 and 23 to provethat thefunction given by
f
(
x
)
x
1 is continuous at each point of its domain.
25 If
f
areall real functions with thesamedomain
A
, and are
all continuous at
x
a
A
, proveby induction that thefunction
f
,
f
,...,
f
f
...
f
, defined by
