<s>
In	O
mathematics	O
,	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
an	O
example	O
of	O
a	O
function	O
that	O
is	O
continuous	O
,	O
but	O
not	O
absolutely	O
continuous	O
.	O
</s>
<s>
It	O
is	O
a	O
notorious	O
counterexample	O
in	O
analysis	O
,	O
because	O
it	O
challenges	O
naive	O
intuitions	O
about	O
continuity	O
,	O
derivative	B-Algorithm
,	O
and	O
measure	O
.	O
</s>
<s>
Though	O
it	O
is	O
continuous	O
everywhere	O
and	O
has	O
zero	O
derivative	B-Algorithm
almost	O
everywhere	O
,	O
its	O
value	O
still	O
goes	O
from	O
0	O
to	O
1	O
as	O
its	O
argument	O
reaches	O
from	O
0	O
to	O
1	O
.	O
</s>
<s>
It	O
is	O
also	O
called	O
the	O
Cantor	B-Algorithm
ternary	I-Algorithm
function	I-Algorithm
,	O
the	O
Lebesgue	O
function	O
,	O
Lebesgue	O
's	O
singular	B-Algorithm
function	I-Algorithm
,	O
the	O
Cantor	O
–	O
Vitali	O
function	O
,	O
the	O
Devil	O
's	O
staircase	O
,	O
the	O
Cantor	B-Algorithm
staircase	I-Algorithm
function	O
,	O
and	O
the	O
Cantor	B-Algorithm
–	I-Algorithm
Lebesgue	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
introduced	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
and	O
mentioned	O
that	O
Scheeffer	O
pointed	O
out	O
that	O
it	O
was	O
a	O
counterexample	O
to	O
an	O
extension	O
of	O
the	O
fundamental	O
theorem	O
of	O
calculus	O
claimed	O
by	O
Harnack	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
was	O
discussed	O
and	O
popularized	O
by	O
,	O
and	O
.	O
</s>
<s>
To	O
define	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
,	O
let	O
be	O
any	O
number	O
in	O
and	O
obtain	O
by	O
the	O
following	O
steps	O
:	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
challenges	O
naive	O
intuitions	O
about	O
continuity	O
and	O
measure	O
;	O
though	O
it	O
is	O
continuous	O
everywhere	O
and	O
has	O
zero	O
derivative	B-Algorithm
almost	O
everywhere	O
,	O
goes	O
from	O
0	O
to	O
1	O
as	O
goes	O
from	O
0	O
to	O
1	O
,	O
and	O
takes	O
on	O
every	O
value	O
in	O
between	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
the	O
most	O
frequently	O
cited	O
example	O
of	O
a	O
real	O
function	O
that	O
is	O
uniformly	O
continuous	O
(	O
precisely	O
,	O
it	O
is	O
Hölder	B-Algorithm
continuous	I-Algorithm
of	O
exponent	O
α	O
=	O
log2/log3	O
)	O
but	O
not	O
absolutely	O
continuous	O
.	O
</s>
<s>
It	O
is	O
constant	O
on	O
intervals	O
of	O
the	O
form	O
(	O
x1x2x3	O
...	O
xn022222	O
,	O
x1x2x3	O
...	O
xn200000	O
)	O
,	O
and	O
every	O
point	O
not	O
in	O
the	O
Cantor	O
set	O
is	O
in	O
one	O
of	O
these	O
intervals	O
,	O
so	O
its	O
derivative	B-Algorithm
is	O
0	O
outside	O
of	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
it	O
has	O
no	O
derivative	B-Algorithm
at	O
any	O
point	O
in	O
an	O
uncountable	O
subset	O
of	O
the	O
Cantor	O
set	O
containing	O
the	O
interval	O
endpoints	O
described	O
above	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
can	O
also	O
be	O
seen	O
as	O
the	O
cumulative	O
probability	O
distribution	O
function	O
of	O
the	O
1/2-1/2	O
Bernoulli	O
measure	O
μ	O
supported	O
on	O
the	O
Cantor	O
set	O
:	O
.	O
</s>
<s>
However	O
,	O
no	O
non-constant	O
part	O
of	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
can	O
be	O
represented	O
as	O
an	O
integral	O
of	O
a	O
probability	O
density	O
function	O
;	O
integrating	O
any	O
putative	O
probability	O
density	O
function	O
that	O
is	O
not	O
almost	O
everywhere	O
zero	O
over	O
any	O
interval	O
will	O
give	O
positive	O
probability	O
to	O
some	O
interval	O
to	O
which	O
this	O
distribution	O
assigns	O
probability	O
zero	O
.	O
</s>
<s>
In	O
particular	O
,	O
as	O
pointed	O
out	O
,	O
the	O
function	O
is	O
not	O
the	O
integral	O
of	O
its	O
derivative	B-Algorithm
even	O
though	O
the	O
derivative	B-Algorithm
exists	O
almost	O
everywhere	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
the	O
standard	O
example	O
of	O
a	O
singular	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
non-decreasing	O
,	O
and	O
so	O
in	O
particular	O
its	O
graph	O
defines	O
a	O
rectifiable	O
curve	O
.	O
</s>
<s>
Because	O
the	O
Lebesgue	O
measure	O
of	O
the	O
uncountably	O
infinite	O
Cantor	O
set	O
is	O
0	O
,	O
for	O
any	O
positive	O
ε	O
<	O
1	O
and	O
δ	O
,	O
there	O
exists	O
a	O
finite	O
sequence	O
of	O
pairwise	O
disjoint	O
sub-intervals	O
with	O
total	O
length	O
<	O
δ	O
over	O
which	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
cumulatively	O
rises	O
more	O
thanε	O
.	O
</s>
<s>
Below	O
we	O
define	O
a	O
sequence	O
 { fn } 	O
of	O
functions	O
on	O
the	O
unit	O
interval	O
that	O
converges	O
to	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
One	O
may	O
check	O
that	O
fn	O
converges	O
pointwise	O
to	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
defined	O
above	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
closely	O
related	O
to	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
the	O
Cantor	O
set	O
is	O
a	O
fractal	B-Application
with	O
(	O
uncountably	O
)	O
infinitely	O
many	O
points	O
(	O
zero-dimensional	O
volume	O
)	O
,	O
but	O
zero	O
length	O
(	O
one-dimensional	O
volume	O
)	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
possesses	O
several	O
symmetries	O
.	O
</s>
<s>
Arbitrary	O
finite-length	O
strings	O
in	O
the	O
letters	O
L	O
and	O
R	O
correspond	O
to	O
the	O
dyadic	O
rationals	O
,	O
in	O
that	O
every	O
dyadic	O
rational	O
can	O
be	O
written	O
as	O
both	O
for	O
integer	O
n	O
and	O
m	O
and	O
as	O
finite	O
length	O
of	O
bits	O
with	O
Thus	O
,	O
every	O
dyadic	O
rational	O
is	O
in	O
one-to-one	O
correspondence	O
with	O
some	O
self-symmetry	O
of	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Writing	O
as	O
a	O
general	O
element	O
of	O
the	O
monoid	O
,	O
there	O
is	O
a	O
corresponding	O
self-symmetry	O
of	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
:	O
</s>
<s>
In	O
fact	O
,	O
a	O
large	O
class	O
of	O
commonly	O
occurring	O
fractals	B-Application
are	O
described	O
by	O
the	O
dyadic	O
monoid	O
;	O
additional	O
examples	O
can	O
be	O
found	O
in	O
the	O
article	O
on	O
de	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
Other	O
fractals	B-Application
possessing	O
self-similarity	O
are	O
described	O
with	O
other	O
kinds	O
of	O
monoids	O
.	O
</s>
<s>
Note	O
that	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
bears	O
more	O
than	O
a	O
passing	O
resemblance	O
to	O
Minkowski	O
's	O
question-mark	O
function	O
.	O
</s>
<s>
This	O
expansion	O
is	O
discussed	O
in	O
greater	O
detail	O
in	O
the	O
article	O
on	O
the	O
dyadic	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
For	O
z	O
=	O
1/3	O
,	O
the	O
inverse	O
of	O
the	O
function	O
x	O
=	O
2C1/3	O
( y	O
)	O
is	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
y	O
=	O
y(x )	O
is	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
for	O
any	O
z1/2	O
,	O
Cz(y )	O
looks	O
like	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
turned	O
on	O
its	O
side	O
,	O
with	O
the	O
width	O
of	O
the	O
steps	O
getting	O
wider	O
as	O
z	O
approaches	O
zero	O
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
also	O
the	O
cumulative	O
distribution	O
function	O
of	O
a	O
measure	O
on	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
Different	O
Cantor	B-Algorithm
functions	I-Algorithm
,	O
or	O
Devil	O
's	O
Staircases	O
,	O
can	O
be	O
obtained	O
by	O
considering	O
different	O
atom-less	O
probability	O
measures	O
supported	O
on	O
the	O
Cantor	O
set	O
or	O
other	O
fractals	B-Application
.	O
</s>
<s>
While	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
has	O
derivative	B-Algorithm
0	O
almost	O
everywhere	O
,	O
current	O
research	O
focusses	O
on	O
the	O
question	O
of	O
the	O
size	O
of	O
the	O
set	O
of	O
points	O
where	O
the	O
upper	O
right	O
derivative	B-Algorithm
is	O
distinct	O
from	O
the	O
lower	O
right	O
derivative	B-Algorithm
,	O
causing	O
the	O
derivative	B-Algorithm
to	O
not	O
exist	O
.	O
</s>
<s>
This	O
analysis	O
of	O
differentiability	O
is	O
usually	O
given	O
in	O
terms	O
of	O
fractal	B-Application
dimension	O
,	O
with	O
the	O
Hausdorff	O
dimension	O
the	O
most	O
popular	O
choice	O
.	O
</s>
<s>
This	O
line	O
of	O
research	O
was	O
started	O
in	O
the	O
1990s	O
by	O
Darst	O
,	O
who	O
showed	O
that	O
the	O
Hausdorff	O
dimension	O
of	O
the	O
set	O
of	O
non-differentiability	O
of	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
the	O
square	O
of	O
the	O
dimension	O
of	O
the	O
Cantor	O
set	O
,	O
.	O
</s>
<s>
Subsequently	O
Falconer	O
showed	O
that	O
this	O
squaring	O
relationship	O
holds	O
for	O
all	O
Ahlfor	O
's	O
regular	O
,	O
singular	O
measures	O
,	O
i.e.Later	O
,	O
Troscheit	O
obtain	O
a	O
more	O
comprehensive	O
picture	O
of	O
the	O
set	O
where	O
the	O
derivative	B-Algorithm
does	O
not	O
exist	O
for	O
more	O
general	O
normalized	O
Gibb	O
's	O
measures	O
supported	O
on	O
self-conformal	O
and	O
self-similar	O
sets	O
.	O
</s>
<s>
Hermann	O
Minkowski	O
's	O
question	O
mark	O
function	O
loosely	O
resembles	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
visually	O
,	O
appearing	O
as	O
a	O
"	O
smoothed	O
out	O
"	O
form	O
of	O
the	O
latter	O
;	O
it	O
can	O
be	O
constructed	O
by	O
passing	O
from	O
a	O
continued	O
fraction	O
expansion	O
to	O
a	O
binary	O
expansion	O
,	O
just	O
as	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
can	O
be	O
constructed	O
by	O
passing	O
from	O
a	O
ternary	O
expansion	O
to	O
a	O
binary	O
expansion	O
.	O
</s>
<s>
The	O
question	O
mark	O
function	O
has	O
the	O
interesting	O
property	O
of	O
having	O
vanishing	O
derivatives	B-Algorithm
at	O
all	O
rational	O
numbers	O
.	O
</s>
