The classical Hartman-Grobman Theorem states that a smooth diffeomorphism
F
(
x
)
F(x)
near its hyperbolic fixed point
x
¯
\bar x
is topological conjugate to its linear part
D
F
(
x
¯
)
DF(\bar x)
by a local homeomorphism
Φ
(
x
)
\Phi (x)
. In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth
F
(
x
)
F(x)
is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a
C
∞
C^\infty
diffeomorphism
F
(
x
)
F(x)
, the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a
C
1
C^1
diffeomorphism
F
(
x
)
F(x)
with
D
F
(
x
)
DF(x)
being
α
\alpha
-Hölder continuous at the fixed point that the local homeomorphism
Φ
(
x
)
\Phi (x)
is differentiable at the fixed point. Here,
α
>
0
\alpha >0
depends on the bands of the spectrum of
F
′
(
x
¯
)
F’(\bar x)
for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on
F
(
x
)
F(x)
cannot be lowered to
C
1
C^1
.