<s>
In	O
mathematics	O
,	O
a	O
piecewise-defined	B-Algorithm
function	I-Algorithm
(	O
also	O
called	O
a	O
piecewise	B-Algorithm
function	I-Algorithm
,	O
a	O
hybrid	O
function	O
,	O
or	O
definition	O
by	O
cases	O
)	O
is	O
a	O
function	O
defined	O
by	O
multiple	O
sub-functions	O
,	O
where	O
each	O
sub-function	O
applies	O
to	O
a	O
different	O
interval	O
in	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
Piecewise	B-Algorithm
definition	O
is	O
actually	O
a	O
way	O
of	O
expressing	O
the	O
function	O
,	O
rather	O
than	O
a	O
characteristic	O
of	O
the	O
function	O
itself	O
.	O
</s>
<s>
A	O
distinct	O
,	O
but	O
related	O
notion	O
is	O
that	O
of	O
a	O
property	O
holding	O
piecewise	B-Algorithm
for	O
a	O
function	O
,	O
used	O
when	O
the	O
domain	B-Algorithm
can	O
be	O
divided	O
into	O
intervals	O
on	O
which	O
the	O
property	O
holds	O
.	O
</s>
<s>
A	O
piecewise	B-Algorithm
linear	O
function	O
(	O
which	O
happens	O
to	O
be	O
also	O
continuous	O
)	O
is	O
depicted	O
as	O
an	O
example	O
.	O
</s>
<s>
Piecewise	B-Algorithm
functions	I-Algorithm
can	O
be	O
defined	O
using	O
the	O
common	O
functional	O
notation	O
,	O
where	O
the	O
body	O
of	O
the	O
function	O
is	O
an	O
array	O
of	O
functions	O
and	O
associated	O
subdomains	O
.	O
</s>
<s>
These	O
subdomains	O
together	O
must	O
cover	O
the	O
whole	O
domain	B-Algorithm
;	O
often	O
it	O
is	O
also	O
required	O
that	O
they	O
are	O
pairwise	O
disjoint	O
,	O
i.e.	O
</s>
<s>
form	O
a	O
partition	O
of	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
In	O
order	O
for	O
the	O
overall	O
function	O
to	O
be	O
called	O
"	O
piecewise	B-Algorithm
"	O
,	O
the	O
subdomains	O
are	O
usually	O
required	O
to	O
be	O
intervals	O
(	O
some	O
may	O
be	O
degenerated	O
intervals	O
,	O
i.e.	O
</s>
<s>
For	O
example	O
,	O
consider	O
the	O
piecewise	B-Algorithm
definition	O
of	O
the	O
absolute	O
value	O
function	O
:	O
</s>
<s>
Here	O
,	O
notice	O
that	O
in	O
order	O
to	O
evaluate	O
a	O
piecewise	B-Algorithm
function	I-Algorithm
at	O
a	O
given	O
input	O
value	O
,	O
the	O
appropriate	O
subdomain	O
needs	O
to	O
be	O
chosen	O
in	O
order	O
to	O
select	O
the	O
correct	O
function	O
—	O
and	O
produce	O
the	O
correct	O
output	O
value	O
.	O
</s>
<s>
A	O
piecewise	B-Algorithm
function	I-Algorithm
is	O
continuous	O
on	O
a	O
given	O
interval	O
in	O
its	O
domain	B-Algorithm
if	O
the	O
following	O
conditions	O
are	O
met	O
:	O
</s>
<s>
The	O
pictured	O
function	O
,	O
for	O
example	O
,	O
is	O
piecewise-continuous	B-Algorithm
throughout	O
its	O
subdomains	O
,	O
but	O
is	O
not	O
continuous	O
on	O
the	O
entire	O
domain	B-Algorithm
,	O
as	O
it	O
contains	O
a	O
jump	O
discontinuity	O
at	O
.	O
</s>
<s>
For	O
a	O
piecewise	B-Algorithm
function	I-Algorithm
to	O
be	O
differentiable	O
on	O
a	O
given	O
interval	O
in	O
its	O
domain	B-Algorithm
,	O
the	O
following	O
conditions	O
have	O
to	O
fulfilled	O
in	O
addition	O
to	O
those	O
for	O
continuity	O
above	O
:	O
</s>
<s>
In	O
applied	O
mathematical	O
analysis	O
,	O
piecewise	B-Algorithm
functions	I-Algorithm
have	O
been	O
found	O
to	O
be	O
consistent	O
with	O
many	O
models	O
of	O
the	O
human	O
visual	O
system	O
,	O
where	O
images	O
are	O
perceived	O
at	O
a	O
first	O
stage	O
as	O
consisting	O
of	O
smooth	O
regions	O
separated	O
by	O
edges	O
.	O
</s>
<s>
In	O
particular	O
,	O
shearlets	B-Algorithm
have	O
been	O
used	O
as	O
a	O
representation	O
system	O
to	O
provide	O
sparse	O
approximations	O
of	O
this	O
model	O
class	O
in	O
2D	O
and	O
3D	O
.	O
</s>
<s>
These	O
are	O
infinitely	O
differentiable	O
,	O
but	O
analyticity	O
holds	O
only	O
piecewise	B-Algorithm
.	O
</s>
<s>
Continuous	O
functions	O
in	O
the	O
reals	O
need	O
not	O
be	O
bounded	O
or	O
uniformly	O
continuous	O
,	O
but	O
are	O
always	O
piecewise	B-Algorithm
bounded	O
and	O
piecewise	B-Algorithm
uniformly	O
continuous	O
.	O
</s>
