Let
A
A
be a finite-dimensional commutative algebra over
R
\mathbb {R}
and let
C
A
r
(
U
)
C_{A}^{r}(U)
,
C
ω
(
U
,
A
)
C^{\omega }(U,A)
and
O
A
(
U
)
\mathcal { O}_{A}(U)
be the ring of
A
A
-differentiable functions of class
C
r
,
0
≤
r
≤
∞
C^{r},\,0 \leq r \leq \infty
, the ring of real analytic mappings with values in
A
A
and the ring of
A
A
-analytic functions, respectively, defined on an open subset
U
U
of
A
n
A^{n}
. We prove two basic results concerning
A
A
-differentiability and
A
A
-analyticity:
1
s
t
1^{st}
)
O
A
(
U
)
=
C
A
∞
(
U
)
⋂
C
ω
(
U
,
A
)
\mathcal { O}_{A}(U) = C^{\infty }_{A}(U) \bigcap C^{\omega }(U,A)
,
2
n
d
2^{nd}
)
O
A
(
U
)
=
C
A
∞
(
U
)
\mathcal { O}_{A}(U) = C^{\infty }_{A}(U)
if and only if
A
A
is defined over
C
\mathbb {C}
.