<s>
there	O
exists	O
a	O
set	O
N	O
of	O
measure	O
0	O
such	O
that	O
for	O
all	O
x	O
outside	O
of	O
N	O
the	O
derivative	B-Algorithm
f(x )	O
exists	O
and	O
is	O
zero	O
,	O
that	O
is	O
,	O
the	O
derivative	B-Algorithm
of	O
f	O
vanishes	O
almost	O
everywhere	O
.	O
</s>
<s>
A	O
standard	O
example	O
of	O
a	O
singular	B-Algorithm
function	I-Algorithm
is	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
,	O
which	O
is	O
sometimes	O
called	O
the	O
devil	O
's	O
staircase	O
(	O
a	O
term	O
also	O
used	O
for	O
singular	B-Algorithm
functions	I-Algorithm
in	O
general	O
)	O
.	O
</s>
<s>
Singular	B-Algorithm
functions	I-Algorithm
occur	O
,	O
for	O
instance	O
,	O
as	O
sequences	O
of	O
spatially	O
modulated	O
phases	O
or	O
structures	O
in	O
solids	O
and	O
magnets	O
,	O
described	O
in	O
a	O
prototypical	O
fashion	O
by	O
the	O
Frenkel	O
–	O
Kontorova	O
model	O
and	O
by	O
the	O
ANNNI	O
model	O
,	O
as	O
well	O
as	O
in	O
some	O
dynamical	O
systems	O
.	O
</s>
<s>
When	O
discussing	O
mathematical	O
analysis	O
in	O
general	O
,	O
or	O
more	O
specifically	O
real	O
analysis	O
or	O
complex	O
analysis	O
or	O
differential	O
equations	O
,	O
it	O
is	O
common	O
for	O
a	O
function	O
which	O
contains	O
a	O
mathematical	O
singularity	O
to	O
be	O
referred	O
to	O
as	O
a	O
'	O
singular	B-Algorithm
function	I-Algorithm
 '	O
.	O
</s>
<s>
A	O
weak	O
derivative	B-Algorithm
is	O
defined	O
that	O
allows	O
singular	B-Algorithm
functions	I-Algorithm
to	O
be	O
used	O
in	O
partial	O
differential	O
equations	O
,	O
etc	O
.	O
</s>
