<s>
The	O
number	O
α	O
is	O
called	O
the	O
exponent	O
of	O
the	O
Hölder	B-Algorithm
condition	I-Algorithm
.	O
</s>
<s>
Hölder	B-Algorithm
spaces	I-Algorithm
consisting	O
of	O
functions	O
satisfying	O
a	O
Hölder	B-Algorithm
condition	I-Algorithm
are	O
basic	O
in	O
areas	O
of	O
functional	B-Application
analysis	I-Application
relevant	O
to	O
solving	O
partial	O
differential	O
equations	O
,	O
and	O
in	O
dynamical	O
systems	O
.	O
</s>
<s>
The	O
Hölder	B-Algorithm
space	I-Algorithm
Ck	O
,	O
α(Ω )	O
,	O
where	O
Ω	O
is	O
an	O
open	O
subset	O
of	O
some	O
Euclidean	O
space	O
and	O
k	O
≥	O
0	O
an	O
integer	O
,	O
consists	O
of	O
those	O
functions	O
on	O
Ω	O
having	O
continuous	O
derivatives	B-Algorithm
up	O
through	O
order	O
k	O
and	O
such	O
that	O
the	O
kth	O
partial	O
derivatives	B-Algorithm
are	O
Hölder	B-Algorithm
continuous	I-Algorithm
with	O
exponent	O
α	O
,	O
where	O
0	O
<	O
α	O
≤	O
1	O
.	O
</s>
<s>
This	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
is	O
finite	O
,	O
then	O
the	O
function	O
f	O
is	O
said	O
to	O
be	O
(	O
uniformly	O
)	O
Hölder	B-Algorithm
continuous	I-Algorithm
with	O
exponent	O
α	O
in	O
Ω	O
.	O
</s>
<s>
If	O
the	O
Hölder	O
coefficient	O
is	O
merely	O
bounded	O
on	O
compact	O
subsets	O
of	O
Ω	O
,	O
then	O
the	O
function	O
f	O
is	O
said	O
to	O
be	O
locally	O
Hölder	B-Algorithm
continuous	I-Algorithm
with	O
exponent	O
α	O
in	O
Ω	O
.	O
</s>
<s>
Let	O
Ω	O
be	O
a	O
bounded	O
subset	O
of	O
some	O
Euclidean	O
space	O
(	O
or	O
more	O
generally	O
,	O
any	O
totally	O
bounded	O
metric	O
space	O
)	O
and	O
let	O
0	O
<	O
α	O
<	O
β	O
≤	O
1	O
two	O
Hölder	B-Algorithm
exponents	I-Algorithm
.	O
</s>
<s>
Then	O
,	O
there	O
is	O
an	O
obvious	O
inclusion	O
map	O
of	O
the	O
corresponding	O
Hölder	B-Algorithm
spaces	I-Algorithm
:	O
</s>
<s>
which	O
is	O
continuous	O
since	O
,	O
by	O
definition	O
of	O
the	O
Hölder	B-Algorithm
norms	I-Algorithm
,	O
we	O
have	O
:	O
</s>
<s>
If	O
0	O
<	O
α	O
≤	O
β	O
≤	O
1	O
then	O
all	O
Hölder	B-Algorithm
continuous	I-Algorithm
functions	I-Algorithm
on	O
a	O
bounded	O
set	O
Ω	O
are	O
also	O
Hölder	B-Algorithm
continuous	I-Algorithm
.	O
</s>
<s>
This	O
also	O
includes	O
β	O
=	O
1	O
and	O
therefore	O
all	O
Lipschitz	O
continuous	O
functions	O
on	O
a	O
bounded	O
set	O
are	O
also	O
C0	O
,	O
α	O
Hölder	B-Algorithm
continuous	I-Algorithm
.	O
</s>
<s>
The	O
function	O
f(x )	O
=	O
xβ	O
(	O
with	O
β	O
≤	O
1	O
)	O
defined	O
on	O
[	O
0	O
,	O
1 ]	O
serves	O
as	O
a	O
prototypical	O
example	O
of	O
a	O
function	O
that	O
is	O
C0	O
,	O
α	O
Hölder	B-Algorithm
continuous	I-Algorithm
for	O
0	O
< α ≤ β, but not for α >	O
β	O
.	O
</s>
<s>
Further	O
,	O
if	O
we	O
defined	O
f	O
analogously	O
on	O
,	O
it	O
would	O
be	O
C0	O
,	O
α	O
Hölder	B-Algorithm
continuous	I-Algorithm
only	O
for	O
α	O
=	O
β	O
.	O
</s>
<s>
If	O
a	O
function	O
is	O
Hölder	B-Algorithm
continuous	I-Algorithm
on	O
an	O
interval	O
and	O
then	O
is	O
constant	O
.	O
</s>
<s>
It	O
does	O
not	O
satisfy	O
a	O
Hölder	B-Algorithm
condition	I-Algorithm
of	O
any	O
order	O
,	O
however	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
is	O
Hölder	B-Algorithm
continuous	I-Algorithm
for	O
any	O
exponent	O
and	O
for	O
no	O
larger	O
one	O
.	O
</s>
<s>
Peano	B-Algorithm
curves	I-Algorithm
from	O
[	O
0	O
,	O
1 ]	O
onto	O
the	O
square	O
[	O
0	O
,	O
1 ]2	O
can	O
be	O
constructed	O
to	O
be	O
1/2Hölder	O
continuous	O
.	O
</s>
<s>
Functions	O
which	O
are	O
locally	O
integrable	O
and	O
whose	O
integrals	O
satisfy	O
an	O
appropriate	O
growth	O
condition	O
are	O
also	O
Hölder	B-Algorithm
continuous	I-Algorithm
.	O
</s>
<s>
then	O
u	O
is	O
Hölder	B-Algorithm
continuous	I-Algorithm
with	O
exponent	O
α	O
.	O
</s>
<s>
Functions	O
whose	O
oscillation	O
decay	O
at	O
a	O
fixed	O
rate	O
with	O
respect	O
to	O
distance	O
are	O
Hölder	B-Algorithm
continuous	I-Algorithm
with	O
an	O
exponent	O
that	O
is	O
determined	O
by	O
the	O
rate	O
of	O
decay	O
.	O
</s>
<s>
for	O
a	O
fixed	O
λ	O
with	O
0	O
<	O
λ	O
<	O
1	O
and	O
all	O
sufficiently	O
small	O
values	O
of	O
r	O
,	O
then	O
u	O
is	O
Hölder	B-Algorithm
continuous	I-Algorithm
.	O
</s>
<s>
Functions	O
in	O
Sobolev	O
space	O
can	O
be	O
embedded	O
into	O
the	O
appropriate	O
Hölder	B-Algorithm
space	I-Algorithm
via	O
Morrey	O
's	O
inequality	O
if	O
the	O
spatial	O
dimension	O
is	O
less	O
than	O
the	O
exponent	O
of	O
the	O
Sobolev	O
space	O
.	O
</s>
<s>
where	O
Thus	O
if	O
u	O
∈	O
W1	O
,	O
p(Rn )	O
,	O
then	O
u	O
is	O
in	O
fact	O
Hölder	B-Algorithm
continuous	I-Algorithm
of	O
exponent	O
γ	O
,	O
after	O
possibly	O
being	O
redefined	O
on	O
a	O
set	O
of	O
measure	O
0	O
.	O
</s>
<s>
Any	O
αHölder	O
function	O
f	O
on	O
a	O
subset	O
X	O
of	O
a	O
normed	O
space	O
E	O
admits	O
a	O
uniformly	O
continuous	O
extension	O
to	O
the	O
whole	O
space	O
,	O
which	O
is	O
Hölder	B-Algorithm
continuous	I-Algorithm
with	O
the	O
same	O
constant	O
C	O
and	O
the	O
same	O
exponent	O
α	O
.	O
</s>
