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A smooth function with certain special properties. Morse functions arise and are used in [[Morse theory|Morse theory]].

Let  $  W $
be a smooth complete (in some Riemannian metric) Hilbert manifold (for example, finite dimensional) whose boundary  $  \partial  W $
is a disconnected union (possibly empty) of manifolds  $  V _ {0} $
and  $  V _ {1} $. 
A Morse function for the triple  $  ( W ;  V _ {0} , V _ {1} ) $
is a smooth (of Fréchet class  $  C  ^ {2} $) 
function  $  f :  W \rightarrow [ a , b ] $, 
$  - \infty < a , b < + \infty $(
or  $  f :  W \rightarrow [ a , \infty ] $
for  $  V _ {1} = \emptyset $), 
such that:

1)  $  f ^ { - 1 } ( a) = V _ {0} $, 
$  f ^ { - 1 } ( b) = V _ {1} $;

2) all critical points (cf. [[Critical point|Critical point]]) of  $  f $
lie in  $  W \setminus  \partial  W = f ^ { - 1 } ( a, b ) $
and are non-degenerate;

3) condition  $  C $
of Palais–Smale is fulfilled (see [[#References|[2]]], [[#References|[3]]]). I.e. on any closed set  $  S \subset  W $
where  $  f $
is bounded and the greatest lower bound of  $  x \rightarrow \| d f ( x) \| $
is zero, there is a critical point of  $  f $.

For example, if  $  f $
is a proper function, that is, all sets  $  f ^ { - 1 } [ c , d ] $, 
$  - \infty < c , d \leq  \infty $, 
are compact (possible only for  $   \mathop{\rm dim}  W < \infty $), 
then  $  F $
satisfies condition  $  C $. 
A Morse function attains a (global) minimum on each connected component of  $  W $. 
If  $  V $
is a finite-dimensional manifold, then for  $  k \geq  2 $
the set of Morse functions of class  $  C  ^ {k} $
is a set of the second category (and, if  $  W $
is compact, even a dense open set) in the space of all functions

$$ 
f :  ( W ;  V _ {0} , V _ {1} )  \rightarrow  ( [ a , b ] , a , b )
$$

in the  $  C  ^ {k} $-
topology.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,   "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.S. Palais,   "Morse theory on Hilbert manifolds"  ''Topology'' , '''2'''  (1963)  pp. 299–340</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Smale,   "Morse theory and a nonlinear generalization of the Dirichlet problem"  ''Ann. of Math.'' , '''80'''  (1964)  pp. 382–396</TD></TR></table>

====Comments====
There exist generalizations to Morse functions on stratified spaces (cf. (the editorial comments to) [[Morse theory|Morse theory]] and [[#References|[a1]]]) and to equivariant Morse functions (cf. [[#References|[a2]]] and [[#References|[a3]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Goreski,   R. MacPherson,   "Stratified Morse theory" , Springer  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Wasserman,   "Morse theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064950/m06495034.png" />-manifolds"  ''Bull. Amer. Math. Soc.'' , '''71'''  (1965)  pp. 384–388</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Wasserman,   "Equivariant differential topology"  ''Topology'' , '''8'''  (1969)  pp. 127–150</TD></TR></table>