Given a certain probability space $(\Omega,\mathcal{A},\mathbb{P})$ and a random variable $X:t\mapsto X(t)$ defined on it, which is the difference between the following statements: $$\color{blue}{\text{every }t}\text{ is }\color{red}{\text{almost surely}}\text{ a nondifferentiability point for }X(t)\tag{1}$$ $$\color{red}{\text{almost surely }} \color{blue}{\text{every }t}\text{ is a nondifferentiability point for }X(t)\tag{2}$$ ?
I would rewrite $(1)$ as: $$\tag{1.int}\forall t\text{, }\mathbb{P}(t\text{ is a nondifferentiability point for}X(t))=1$$ and $(2)$ as: $$\tag{2.int}\mathbb{P}(t\text{ is a nondifferentiability point for}X(t), \forall t)=1$$
First, I don't know whether $(1.\text{int})$ and $(2.\text{int})$ are correct "rewritings" of $(1)$ and $(2)$ (resp.).
In general, whichever the difference between $(1)$ and $(2)$, which is the gist of such a difference from a mathematical standpoint? I cannot grasp it.
Could you please give an example of a random variable for which $(1)$ holds true, but $(2)$ does NOT hold true? (or viceversa)
On $(0,1)$ with Lebesgue measure let $X_t(\omega)=|t-\omega|$. Then $P(X_t \, \text {is differentiable at } \, t)=1$ for each $t$ and $P(X_t \, \text {is differentiable at every point} \, t)=0$.
(2) implies (1) always.