Quiz, week 5
Is
y
n
=
1
y_n = 1
a subsequence of
x
n
=
n
x_n = n
x
n
=
1
x_n = 1
x
n
=
(
−
1
)
n
x_n = (-1)^n
Which of these functions are continuous?
f
(
x
)
=
1
/
x
f(x) = 1/x
,
f
:
ℝ
∖
{
0
}
→
ℝ
f : \mathbb{R} \backslash \{0\} \rightarrow \mathbb{R}
f
:
ℝ
→
ℝ
f : \mathbb{R} \rightarrow \mathbb{R}
where
f
(
x
)
=
{
1
x
if
x
≠
0
,
0
if
x
=
0
.
f(x) = \begin{cases} \tfrac{1}{x} & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases}
u
:
X
→
ℝ
u : X \rightarrow \mathbb{R}
where
X
=
{
attack
,
retreat
}
X = \{\text{attack}, \text{retreat}\}
and
u
(
x
)
=
{
1
if
x
=
attack,
0
if
x
=
retreat.
u(x) = \begin{cases} 1 & \text{if $x = $ attack,} \\ 0 & \text{if $x = $ retreat.} \end{cases}
f
(
x
)
=
x
f(x) = x
where
f
:
ℕ
→
ℝ
f : \mathbb{N} \rightarrow \mathbb{R}
f
(
x
)
=
I
(
x
>
0
)
f(x) = I(x > 0)
where
f
:
{
1
n
:
n
∈
ℕ
}
∪
{
0
}
→
ℝ
f : \{\tfrac{1}{n} : n \in \mathbb{N}\} \cup \{0\} \rightarrow \mathbb{R}
.
Which of the following sets are open in
(
[
0
,
1
]
,
d
2
)
([0, 1], d_2)
?
(
0
,
1
)
(0, 1)
[
0
,
1
]
[0, 1]
[
0
,
1
)
[0, 1)
[
0
,
1
2
]
[0, \tfrac{1}{2}]
[
0
,
1
2
)
[0, \tfrac{1}{2})
Consider
f
(
x
)
=
x
2
f(x) = x^2
,
f
:
ℝ
→
ℝ
f : \mathbb{R} \rightarrow \mathbb{R}
. What is:
f
−
1
(
1
)
f^{-1}(1)
f
−
1
(
[
0
,
1
]
)
f^{-1}([0, 1])
f
−
1
(
(
0
,
1
)
)
f^{-1}((0, 1))