The Wiener process is defined by the three properties:
1. $W(0) = 0$, 
2. $W(t)$ is almost surely continuous, and 
3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s < t$).


What would be an example of a process which satisfies 1) and 3), but not 2) ?


I am going to teach an introductory class on Brownian motion at advanced undergrad level.
Just wanted to make sure that all the conditions are mutually independent.



Wiener process related counterexample
This is not hard to find such an example. Let $P$ be Wiener measure on the space $\Omega = C([0,\infty))$ of continuous functions $t\mapsto \omega(t)$. Then the process $\omega(t)$ satisfies all three conditions of a Brownian motion.


Now let's define a new process $W(t)$ that is "almost" equal to $\omega(t)$, but where we deliberately wreck the sample path continuity. 


Take any random time $T:\Omega\to [0,\infty)$ that has a continuous distribution on $(\Omega, P)$,
and let $W(t,\omega)=\omega(t)$ when $t\not=T(\omega)$, but $W(t,\omega)=\omega(t)+1$ otherwise. The process $W(t)$ still satisfies 1 and 3 but the sample path continuity fails at exactly at the time point $T(\omega)$ for each $\omega$. 


There are many such random times $T$, for example you could use $T(\omega):=\inf [t>0: \omega(t)=1 ]$, i.e. the hitting time of 1.


