Let
F
:
X
→
Y
F:X \to Y
be a
C
1
{C^1}
Fredholm map of index zero between two Banach spaces. Defining the singular set
B
=
{
x
|
F
′
(
x
)
B = \{ x|F’(x)
is not surjective}, we study the local and global effect of B on the map F. In particular it is shown that if
b
∈
B
b \in B
is isolated in B, then, for
dim
X
\dim X
and
dim
Y
⩾
3
\dim Y \geqslant 3
, F is a local homeomorphism at b. We then show that if B consists of discrete points, F is a global homeomorphism of X onto Y. A nonlinear partial differential equation is included as an illustration.