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Lebl-Basic-Analysis-I
· Problem 3.4.10
Continuous Functions
Mathematics · uniform-continuity
Problem
a) Find a continuous
f
:
(
0
,
1
)
→
R
f : (0,1) \to \mathbb{R}
f
:
(
0
,
1
)
→
R
and a sequence
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
in
(
0
,
1
)
(0,1)
(
0
,
1
)
that is Cauchy, but such that
{
f
(
x
n
)
}
n
=
1
∞
\{f(x_n)\}_{n=1}^\infty
{
f
(
x
n
)
}
n
=
1
∞
is not Cauchy. b) Prove that if
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
is continuous, and
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
is Cauchy, then
{
f
(
x
n
)
}
n
=
1
∞
\{f(x_n)\}_{n=1}^\infty
{
f
(
x
n
)
}
n
=
1
∞
is Cauchy.
Related theory
uniform-continuity
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