TITLE: Local definition of Hölder continuity
QUESTION [7 upvotes]: 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

REPLY [9 votes]: 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 $U \ni x_0$.