Local definition of Hölder continuity

Question

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

Answer 1

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 $U \ni x_0$.

Answer 2

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.