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Lebl-Basic-Analysis-I
· Problem 4.1.12
The Derivative
Mathematics · differentiability-squeeze
Problem
Suppose
f
:
I
→
R
f : I \to \mathbb{R}
f
:
I
→
R
,
g
:
I
→
R
g : I \to \mathbb{R}
g
:
I
→
R
, and
h
:
I
→
R
h : I \to \mathbb{R}
h
:
I
→
R
, are functions. Suppose
c
∈
I
c \in I
c
∈
I
is such that
f
(
c
)
=
g
(
c
)
=
h
(
c
)
f(c) = g(c) = h(c)
f
(
c
)
=
g
(
c
)
=
h
(
c
)
,
g
g
g
and
h
h
h
are differentiable at
c
c
c
, and
g
′
(
c
)
=
h
′
(
c
)
g'(c) = h'(c)
g
′
(
c
)
=
h
′
(
c
)
. Furthermore, suppose
h
(
x
)
≤
f
(
x
)
≤
g
(
x
)
h(x) \leq f(x) \leq g(x)
h
(
x
)
≤
f
(
x
)
≤
g
(
x
)
for all
x
∈
I
x \in I
x
∈
I
. Prove
f
f
f
is differentiable at
c
c
c
and
f
′
(
c
)
=
g
′
(
c
)
=
h
′
(
c
)
f'(c) = g'(c) = h'(c)
f
′
(
c
)
=
g
′
(
c
)
=
h
′
(
c
)
.
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