Is there any analysis method of non-differentiability?

There is Rademacher's theorem which says that a function that is Lipschitz continuous is differentiable (almost everywhere). So one way of interpreting your question is to "how close to Lipschitz continuous is a function". Lipschitz continuity is a specific case of Hölder continuity.

A function is $\alpha$-Hölder continuous for $\alpha\in (0,1]$ if

$$|f(x)-f(y)|<K|x-y|^\alpha$$

If $\alpha=1$ then we say the function is Lipschitz.

Note that if $f$ is $\alpha$-Hölder, then it is $\beta$-Hölder for $\beta<\alpha$.

So if $f$ is $.6$-Hölder and $g$ is $.7$-Hölder, then $g$ is "closer" to being Lipschitz, i.e. differentiable.

Here are some pictures of various functions that are of increasing Hölder continuity:

Hölder continuous for all $\alpha \in(0,0.15)$:

Hölder continuous for all $\alpha \in (0,0.55)$:

Hölder continuous for all $\alpha\in (0,0.75)$:

Hölder continuous for all $\alpha\in (0,0.95)$:

Note how the paths get "nicer"/closer to being able to differentiate.