Let $(N^{(n)})_{n\in \Bbb{Z}}$ be a sequence of i.i.d. Poisson processes. Then define $Y_t=\sum_{n=0}^\infty 4^{-n} N^{(n)}_{3^nt}$ I want to show that almost surely for all intervals $(u,v)\subset [0,\infty)$, $Y:=(Y_t)_{t\geq 0}$ is not continuous.
I could prove that if we fix $t$ then $Y$ is continuous at $t$. But not I need to show that on every interval it is not.
But unfortunately I have no idea how to construct a situation on $(u,v)$ which tells me that it is not continuous there.
Can someone help me further?
Hint: For each $(u,v)$, $Y_t$ would be continuous on $(u,v)$ if and only if none of the Poisson processes $N^{(n)}_{3^n t}$ had a jump in that interval. For each $n$, compute the probability of the event that $N^{(n)}_{3^n t}$ has a jump in $(u,v)$. Then use independence to show the probability that at least one such event happens is 1. The second Borel-Cantelli lemma could help (you could show the stronger statement that almost surely, infinitely many such events happen).
We are not quite done yet: we have that for each $(u,v)$ the process is a.s. not continuous on $(u,v)$, but the null set where this fails could depend on the interval $(u,v)$. We want to show that almost surely, the process is continuous on no $(u,v)$; that is, we want a single null set that works for all $(u,v)$. To get this, note that it suffices to show that a.s, the process is continuous on no rational interval $(u,v)$, and that there are only countably many such intervals.