This entry devotes the following fundamental question: if is Hölder continuous, then how about for some constant ? Throughout this entry, we work on which is not necessarily bounded.

Firstly, we have an elementary result

Proposition. If and are -Hölder continuous and bounded, so is .

*Proof*. The proof is simple, we just observe that

which yields

.

Consequently,

for any positive integer number and any -Hölder continuous and bounded function , function is also -Hölder continuous and bounded.

Let us assume , is -Hölder continuous and bounded, is a constant. Let . Since , we may assume is also bounded away from zero, that means there exist two constants such that

.

We now study the -Hölder continuity of . Observe that function

is sub-additive in the sense that

.