Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].v
Requirements: