Graphics Reference
In-Depth Information
A function
f
is defined on
R
by
f
(
x
)
x
sin 1/
x
when
x
0,
f
(0)
0.
Assuming that thesinefunction is continuous on R and that
sin
x
1 for all values of
x
, provethat
f
is continuous on R.
(Hint: useqns 48, 22 and 26 for
x
0, and qns 32, 26, and 36 for
x
0.)
50 Let
f
(
x
)
(
x
2)(
x
3) be defined on R and let
r
denote the same
function as in qn 47.
What is themaximum domain of definition of
r
f
? Is this function
continuous throughout its domain? Examinethegraphs of
f
and
r
f
on a computer or a graphics calculator.
51 Let
f
(
x
)
a
x
, be defined on R.Let
r
denote the same function as in qn 47. What is themaximum
domain of definition of
r
f
? Is this function continuous throughout
its domain?
a
x
a
x
...
a
52 Let
f
:
A
R bea real function which is continuous at
x
a
, with
f
(
a
)
0. Let
r
denote the same function as in qn 47. Prove that
r
f
is continuous at
x
a
.
Thefunction
r
f
is also commonly denoted by 1/
f
. Givea dirct
proof that 1/
f
is continuous at
a
using qns 3.65 and 3.66, the
reciprocal rule.
53 Provethat thefunction given by
f
(
x
)
1) is
continuous on R. Examineits graph on a computr or a graphics
calculator.
(
x
x
)/(
x
54 Let
f
:
A
R and
g
:
A
R bereal functions which areboth
continuous at
a
.Let
G
x
g
(
x
)
0
.If
g
(
a
)
0, provethat the
function
f
/
g
:
A
G
R defined by (
f
/
g
)(
x
)
f
(
x
)/
g
(
x
) is continuous
at
a
. Either consider
f
· (
r
g
) or use qn 3.67, the quotient rule,
directly.
55 (
Cauchy
, 1821) Thefunction
f
: R
R has theproprty that
f
(
x
y
)
f
(
x
)
f
(
y
) for all
x
,
y
R
.
(i) Provethat
f
(0)
0.
(ii) Provethat
f
(
f
(
x
).
(iii) If
f
is continuous at
x
0, provethat
f
is continuous on R.
(iv) If
f
(1)
a
, provethat
f
(
n
)
an
, for
n
Z.
If
n
N, show that
n
·
f
(
x
/
n
)
f
(
x
).
(v) If
f
(1)
a
, provethat
f
(
p
/
q
)
ap
/
q
, for
p
,
q
Z,
q
0.
(vi) If
f
(1)
x
)
a
and
f
is continuous on
R
, provethat
f
(
x
)
ax
.
