Let
∅
≠ D
⊆
R. D is said to be
convex
if
2200
a,b
∈
D, the line segment connecting
and
is contained in D.
Let
∅
≠ D
⊆
R. D is said to be
connected
if
2200
,
∈
D,
5
φ
: [0,1]
→
D such that
φ
(0)
=
,
φ
(1) =
Let D ≠
∅
.
, x
0
∈
D. f is
continuous
at x
0
if
2200
{a
n
}
→
x
0
, a
n
∈
D,
{f(x
n
)}
→
f(x
0
).
Let D ≠
∅
.
, x
0
∈
D. f is
continuous
at x
0
if
2200ε
> 0,
5δ
such that if  x – x
0
 <
δ
, x
∈
D, f(x) – f(x
0
) <
ε
Let
∅
≠ D
⊆
R.
.
is said to be
uniformly continuous
on D if
2200
{u
n
},
{v
n
}
∈
D if {u
n
 v
n
}
→
0, then 
(u
n
) –
(v
n
) 
→
0
Let
∅
≠ D
⊆
R.
.
is said to be
uniformly continuous
on D if
2200ε
> 0,
5δ
such that if  x – y  <
δ
, x,y
∈
D, f(x) – f(y) <
ε
Let O
⊆
R. O is said to be
open
if for any r
∈
O,
5ε
> 0 such that (r
ε
, r+
ε
)
⊆
O
Let D
⊆
R. O
i
, i
∈
I, be open. If D
⊆
, then
is an
open cover
of D.
Let
∅
≠ D
⊆
R. x
0
∈
D. We say x
0
is an
isolated point
of D if
5ε
>0 such that
(x
0

ε
, x
0
+
ε
)
∩
D = { x
0
}
Let
∅
≠ D
⊆
R.
.
is said to be
monotonically increasing
if for u , v
∈
D, u
≤ v,
(u) ≤
(v)
Let
∅
≠ D
⊆
R.
.
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 Spring '10
 Roberts
 Topology, Continuous function, Order theory, limit point, U. Let

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