The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class
C
m
{C^m}
. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class
C
m
{C^m}
or
C
∞
{C^\infty }
. Such a property is characteristic for the solutions
f
:
R
n
→
R
f:{R^n} \to R
of the equations
(
∗
)
∑
i
=
1
k
a
i
(
x
,
t
)
f
(
ϕ
i
(
x
,
t
)
)
=
F
(
x
,
f
(
λ
1
(
x
)
)
,
…
,
f
(
λ
s
(
x
)
)
)
+
b
(
x
,
t
)
,
\begin{equation}\tag {$\ast $} \sum \limits _{i = 1}^k {{a_i}(x,t)f({\phi _i}(x,t)) = F(x,f({\lambda _1}(x)), \ldots ,f({\lambda _s}(x))) + b(x,t),} \end{equation}
where
x
∈
R
n
,
t
∈
R
r
,
n
⩾
1
,
r
⩾
1
x \in {R^n},t \in {R^r},n \geqslant 1,r \geqslant 1
and where the functions
ϕ
i
:
R
n
+
r
→
R
n
,
λ
j
:
R
n
→
R
n
,
a
i
:
R
n
+
r
→
R
,
b
:
R
n
+
r
→
R
,
F
:
R
n
+
s
→
R
{\phi _i}:{R^{n + r}} \to {R^n},{\lambda _j}:{R^n} \to {R^n},{a_i}:{R^{n + r}} \to R,b:{R^{n + r}} \to R,F:{R^{n + s}} \to R
satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating
(
∗
)
(\ast )
and fixing t is of constant strength, hypoelliptic at a point
x
0
{x_0}
. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of
(
∗
)
(\ast )
, is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.