What does Dini continuity mean? What does Dini continuity (the integral condition) mean visually?
Description of Dini contuity:
https://en.wikipedia.org/wiki/Dini_continuity
 A: Start with this visualization of Lipschitz continuity by A. di M.:

As you slide the focus of the green overlay along the graph of the function, the rest of the graph stays within the green area. The Lipschitz condition says that the modulus of continuity is bounded by a linear function, so you can use a green area with straight sides.
The Hölder condition says that the modulus of continuity is bounded by a power law like $\sqrt t$, so you can use an area with curved sides that become vertical where they meet at the focus. This condition is easier to satisfy than the Lipschitz condition, because the green area is larger.
The Dini condition allows certain moduli of continuity that are even closer to vertical than a power law, such as $(\log t)^{-2}$. So in the picture, you can imagine a very sharp pinch in the middle.
A: The Dini condition has no full characterization because there is no borderline case of $\omega$ where the integral converges. This condition arises out of the Dini convergence criterion for the Fourier series $S_{f}(x)$ of a function $f$ at $x$. You can write the difference between the truncated Fourier series $S_{f}^{N}(x)$ for $f$ on $[-\pi,\pi]$ and some potential limit $L$ as
$$
   S_{f}^{N}(x)-L = \frac{1}{\pi}\int_{0}^{\pi}\frac{f(\theta+\theta')+f(\theta-\theta')-2L}{\sin((\theta-\theta')/2)}\sin((N+\frac{1}{2})(\theta-\theta'))d\theta'.
$$
If $\theta$ is fixed and $\theta'=\theta+h$, then
$$
          \frac{f(\theta+\theta')+f(\theta-\theta')-2L}{\theta-\theta'}
$$
has the form $\omega(h)/h$. If this is absolutely integrable in a neighborhood of $0$, then then the Riemann-Lebesgue lemma gives $\lim_{N} (S_{f}^{N}(x)-L)=0$, meaning that the Fourier series for $f$ converges to $L$ at $\theta$.
