Let $E$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E$ be càdlàg with $x(0)=0$. Moreover, let $x(0-):=0$, $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t>0$$ and $$\Delta x(t):=x(t)-x(t-)\;\;\;\text{for }t\ge0.$$
Question 1: Does $[0,\infty)\ni t\mapsto\Delta x(t)$ admit any kind of regularity (maybe some weaker notion of continuity)?
If I think, for example, of $x$ as being a path of a Poisson process, it doesn't seem to be regular in any sense, but I might miss something.
Question 2: If there is a $c>0$ such that $\left\|\Delta x(t)\right\|_E\le c$ for all $t\ge0$ and $$\tau:=\inf\{t\ge 0:\|x(t)\|_E\ge c\},$$ how do we see that for all $t\ge 0$ the inequality $\|x(t)\|_E\ge2c$ implies $\tau<t$?
Intuitively, this seems to be trivial, since $\|x(t)\|_E\ge2c$ should imply that a jump must have been occurred before time $t$, but how do we need to argue rigorously?
Question 3: On reason why I've asked question 1 is the following: If $t>0$, is $x(t-)=\lim_{\substack{s\to t-\\s\in\mathbb Q}}x(s)$? If not, how do we see that when $(\Omega,\mathcal A)$ is a measurable space and $X:\Omega\times[0,\infty)\to E$ is measurable and $X(\omega)$ is càdlàg for all $\omega\in\Omega$, that $\Delta X$ is measurable as well?
For question 2, I don't think this is true as written. If we let $x(t) := t 1_{t < 1} + 21_{t \ge 1},$ $|\Delta x(t)| \le 1$ for all $t$ and $x(1) \ge 2$, but $\inf\{t \ge 0 : |x(t)| \ge 1\} = 1$ so $\tau \not < 1$.
If we change it to $\|x(t)\| > 2c,$ then it does hold. We have $x(t) = x(t-) + \Delta x(t)$, so $$2c < \|x(t)\| = \|x(t-)+\Delta x(t)\| \le \|x(t-)\| + \|\Delta x(t)\| \le \|x(t-)\| + c.$$ Therefore $c < \|x(t-)\|$, and hence $c < \lim_{s \rightarrow t-} \|x(s)\|$. This implies there exists $s < c$ with $\|x(s)\| > c$, so $\tau \le s < t$.