Graphics Reference
In-Depth Information
a
a
a
21 Let
a
...
.
A function
f
:[
a
,
a
]
R is defined by
f
(
x
)
x
a
x
a
...
x
a
.Is
f
continuous? Is
f
bounded? Where does it attain its bounds?
A function
g
:[
a
,
a
]
R
is defined by
g
(
x
)
(
x
a
)
(
x
a
)
...
(
x
a
)
.Is
g
continuous? Is
g
bounded? Where does it attain its bounds?
22
The waterfall function
If thefunction
f
:[
a
,
b
]
is continuous at every point of its
domain, must thefunction
g
:[
a
,
b
]
R defined by
g
(
t
)
R
f
(
x
)
a
x
t
sup
be continuous at every point?
23 A function
f
is defined on [
a
,
b
].
x
a
i
(
b
a
)/
n
. Under what
conditions will the sequence with
n
th term (
f
(
x
))/
n
be
convergent as
n
?
24 (
Newton—Raphson
) Under what circumstances will a sequence (
a
)
defined by the relation
a
a
f
(
a
)/
f
(
a
) converge? Try
applying the Second Mean Value Theorem.
25 (
Cauchy
, 1821) What properties can be established for a function
f
:
f
(
x
)·
f
(
y
)? What further properties can
you claim if
f
is continuous with
f
(1)
2?
R
R
such that
f
(
x
y
)
26 For each of the following functions
f
: (0, 1]
R, determine whether
(i) lim
f
(
x
) exists, (ii) whether
f
can be extended to a continuous
function at 0, (iii) whether
f
can be extended to a differentiable
function at 0.
f
(
x
)
x
[1/
x
],
x
[1/
x
],
x
[1/
x
] and
x
[1/
x
].
