How does pointwise Hölder continuous on compact subsets not imply locally Hölder continuous?

Question

When Gilbarg and Trudinger introduced the Hölder spaces, they mentioned on page 52 that

Furthermore note that local Hölder continuity is a stronger property than pointwise Hölder continuity in compact subsets.

without further elaboration.

Can someone kindly explain why?

Answer 1

It is easiest to illustrate this for pointwise $\alpha$-Holder functions with $\alpha=1$, i.e., pointwise Lipschitz functions. Consider $$f(x)=x^2\sin(1/x^2)$$ for $x$ in $[-1,1] \setminus\{0\}$, with $f(0)=0$. The pointwise Lipschitz constant in this case is simply $|f'(x)|$, which is well defined and finite everywhere in $[-1,1]$, but is not bounded in any neighborhood of $0$.