Graphics Reference
In-Depth Information
(
x
x
34 Sketch the graph of the real function given by
f
(
x
)
),
and provethat it is continuous on R.
35 Wedefinea function
f
: R
R by
f
(
x
)
0 when
x
is irrational,
f
(
x
)
x
when
x
is rational.
Provethat, for all
x
(
x
x
f
(
x
)
(
x
x
)
).
Locatethegraph of thefunction
f
in relation to those you have
drawn for qns 33 and 34.
Is there a value of
x
for which
(
x
x
)
(
x
x
)?
A squeeze rule
36 If real functions
f
,
g
, and
h
havethesamedomain
A
, and
(i)
f
(
x
)
g
(
x
)
h
(
x
), for all
x
A
,
(ii)
f
and
h
arecontinuous at
x
a
A
,
(iii)
f
(
a
)
h
(
a
),
use qn 3.54(viii), the squeeze rule for convergent sequences, to prove
that
g
is continuous at
x
a
.
37 Useqn 36 to provethat thefunction
f
of qn 35 is continuous at
x
0, and usearguments likethoseof qn 20 to provethat this
function is not continuous at any other point in its domain.
This example highlights an important consequence of our definition
of continuity sincethepossibility of continuity at a singleisolated
point does not spring from an intuitive view of the concept.
Continuityof composite functions and quotients of continuous
functions
38 If
f
: R
R is defined by
f
(
x
)
sin
x
and
g
: R
R is defined by
g
(
x
)
, what are
f
(
g
(
x
)) and
g
(
f
(
x
))?
Look at thegraphs of thsetwo compositefunctions on a
computer screen.
Generally, if
f
and
g
aretwo real functions and thedomain of
f
contains therangeof
g
, the
composite function f
g
is defined on the
domain of
g
by (
f
g
)(
x
)
f
(
g
(
x
)).
Noticethat thecompositefunction
f
g
is generally different from
thecompositefunction
g
f
.
x
