<s>
In	O
mathematics	O
,	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
is	O
a	O
self-affine	O
curve	O
constructible	O
by	O
midpoint	O
subdivision	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
as	O
the	O
Takagi	B-Algorithm
curve	I-Algorithm
,	O
after	O
Teiji	O
Takagi	O
who	O
described	O
it	O
in	O
1901	O
,	O
or	O
as	O
the	O
Takagi	O
–	O
Landsberg	O
curve	O
,	O
a	O
generalization	O
of	O
the	O
curve	O
named	O
after	O
Takagi	O
and	O
Georg	O
Landsberg	O
.	O
</s>
<s>
It	O
is	O
a	O
special	O
case	O
of	O
the	O
more	O
general	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
;	O
see	O
also	O
fractal	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
for	O
a	O
parameter	O
;	O
thus	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
is	O
the	O
case	O
.	O
</s>
<s>
The	O
value	O
is	O
known	O
as	O
the	O
Hurst	B-Algorithm
parameter	I-Algorithm
.	O
</s>
<s>
The	O
blancmange	B-Algorithm
curve	I-Algorithm
can	O
be	O
visually	O
built	O
up	O
out	O
of	O
triangle	O
wave	O
functions	O
if	O
the	O
infinite	O
sum	O
is	O
approximated	O
by	O
finite	O
sums	O
of	O
the	O
first	O
few	O
terms	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Takagi	B-Algorithm
curve	I-Algorithm
of	O
parameter	O
is	O
defined	O
on	O
the	O
unit	O
interval	O
(	O
or	O
)	O
if	O
.	O
</s>
<s>
The	O
Takagi	B-Algorithm
function	I-Algorithm
of	O
parameter	O
is	O
continuous	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
functions	O
defined	O
by	O
the	O
partial	O
sums	O
are	O
continuous	O
and	O
converge	B-Algorithm
uniformly	I-Algorithm
toward	O
,	O
since	O
:	O
</s>
<s>
This	O
value	O
can	O
be	O
made	O
as	O
small	O
as	O
we	O
want	O
by	O
selecting	O
a	O
big	O
enough	O
value	O
of	O
n	O
.	O
Therefore	O
,	O
by	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
theorem	I-Algorithm
,	O
is	O
continuous	O
if	O
|w|	O
<	O
1	O
.	O
</s>
<s>
Since	O
the	O
absolute	O
value	O
is	O
a	O
subadditive	O
function	O
so	O
is	O
the	O
function	O
,	O
and	O
its	O
dilations	O
;	O
since	O
positive	O
linear	O
combinations	O
and	O
point-wise	O
limits	O
of	O
subadditive	O
functions	O
are	O
subadditive	O
,	O
the	O
Takagi	B-Algorithm
function	I-Algorithm
is	O
subadditive	O
for	O
any	O
value	O
of	O
the	O
parameter	O
.	O
</s>
<s>
For	O
,	O
one	O
obtains	O
the	O
parabola	O
:	O
the	O
construction	O
of	O
the	O
parabola	O
by	O
midpoint	O
subdivision	O
was	O
described	O
by	O
Archimedes	B-Device
.	O
</s>
<s>
For	O
values	O
of	O
the	O
parameter	O
the	O
Takagi	B-Algorithm
function	I-Algorithm
is	O
differentiable	O
in	O
classical	O
sense	O
at	O
any	O
which	O
is	O
not	O
a	O
dyadic	O
rational	O
.	O
</s>
<s>
For	O
the	O
blancmange	B-Algorithm
function	I-Algorithm
it	O
is	O
of	O
bounded	O
variation	O
on	O
no	O
non-empty	O
open	O
set	O
;	O
it	O
is	O
not	O
even	O
locally	O
Lipschitz	O
,	O
but	O
it	O
is	O
quasi-Lipschitz	O
,	O
indeed	O
,	O
it	O
admits	O
the	O
function	O
as	O
a	O
modulus	O
of	O
continuity	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
action	O
of	O
a	O
general	O
element	O
maps	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
on	O
the	O
unit	O
interval	O
 [ 0 , 1 ] 	O
to	O
a	O
sub-interval	O
for	O
some	O
integers	O
m	O
,	O
n	O
,	O
p	O
.	O
The	O
mapping	O
is	O
given	O
exactly	O
by	O
where	O
the	O
values	O
of	O
a	O
,	O
b	O
and	O
c	O
can	O
be	O
obtained	O
directly	O
by	O
multiplying	O
out	O
the	O
above	O
matrices	O
.	O
</s>
<s>
The	O
above	O
three-dimensional	O
representation	O
is	O
just	O
one	O
of	O
many	O
representations	O
it	O
can	O
have	O
;	O
it	O
shows	O
that	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
is	O
one	O
possible	O
realization	O
of	O
the	O
action	O
.	O
</s>
<s>
That	O
is	O
,	O
there	O
are	O
representations	O
for	O
any	O
dimension	O
,	O
not	O
just	O
3	O
;	O
some	O
of	O
these	O
give	O
the	O
de	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
The	O
definite	B-Algorithm
integral	I-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
Repeated	O
integrals	B-Algorithm
transform	O
under	O
a	O
5	O
,	O
6	O
,...	O
dimensional	O
representation	O
.	O
</s>
<s>
(	O
suitably	O
normalized	O
)	O
approaches	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
.	O
</s>
