Local definition of Hölder continuity

What does it mean for a continuous function $f$ on $\mathbb{R}$ to be Hölder continuous with exponent $\alpha$ at a point $x_0$ ?

I only now the global definition: A function $f$ on $\mathbb{R}$ is (globally) Hölder continuous with exponent $\alpha$ if

$\sup_{x \neq y} \frac{| f(x) - f(y) |}{ |x - y|^\alpha} < + \infty$

Thanks for the clarification!

Regards, Si

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Locally means that we can find a neighborhood $V$ of $x_0$ such that $\sup\{\frac{|f(x)-f(y)|}{|x-y|},x,y\in V,x\neq y\}$ is finite. – Davide Giraudo May 2 '12 at 11:01
@DavideGiraudo: Hmmm... So if one says that Brownian motion is $P$-a.s. nowhere Hölder continuous with exponent $\alpha > 1/2$, is the meaning that $P$-a.s. no point $x_0$ has a neighborhood $V$ such that $\sup_{x, y \in V, x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha} < + \infty$ or that no point $x_0$ has a neighborhood $V$ such that $\sup_{x \in V, x \neq x_0} \frac{|f(x) - f(x_0)|}{|x - x_0|^\alpha} < + \infty$ ? Thanks a lot Davide! – Mad Si May 2 '12 at 13:34

as far as I remember, one calls $f$

• Hölder continuous of exponent $\alpha$ iff $\sup_{x,y\in\mathbb R} \frac{|f(x) - f(y)|}{|x-y|^\alpha} <\infty$
• locally Hölder continuous of exponent $\alpha$ iff $\sup_{x,y\in K} \frac{|f(x) - f(y)|}{|x-y|^\alpha} <\infty$ for each compact $K \subset \mathbb R$
• Hölder continuous at $x_0$ of exponent $\alpha$ iff $\sup_{x\in U} \frac{|f(x) - f(x_0)|}{|x-x_0|^\alpha} <\infty$ for some neighbourhood $U \ni x_0$.
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Hence Hölder continuity at $x_0$ means exactly $f(x) = O(|x|^\alpha)$? – Siminore May 2 '12 at 13:45
As far as I remember, yes, that's Hölder continuity at 0, for general $x_0$ we should write $f(x) = O(|x-x_0|^\alpha)$, $x \to x_0$.# – martini May 2 '12 at 13:47
I think your suggestion is wrong: we should write $f(x)=f(x_0)+O(|x-x_0|^\alpha)$. Anyway, it's interesting, I had never studied Hölder continuity at a precise point. – Siminore May 2 '12 at 13:54
Oh, of course you're right ... – martini May 2 '12 at 14:01
Yes, these three definitions agree with those given in standard texts such as Gilbarg and Trudinger Elliptic Partial Differential Equations of Second Order, p.52 – Willie Wong May 2 '12 at 14:12

As usual, the term "local" (or "locally") means that the definition should be restricted to any neighborhood. In you case, $f$ is locally Hölder continuous if, for every interval $(a,b)$ there exists a constant $C>0$ such that $|f(x)-f(y)| \leq C |x-y|^\alpha$ for every $x$, $y \in [a,b]$. Clearly enough, this amounts to considering $f_{|[a,b]}$ instead of $f$ in the global definition. Notice that the constant $C$ depends on $(a,b)$, so the local definition is, in general, strictly different than the global one.

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