I'm taking an undergraduate course that introduces stochastic processes and is finishing with Brownian motion. The discussion points out that a standard Brownian motion $\{B_t: t\geq0\}$:
- has independent increments on disjoint intervals (OK, independence often leads to factorization, which is useful)
- for non-negative $t$, $B_t \sim N(0, t)$ (OK, the normal distribution is familiar)
- has continuous sample paths.
I find the last property the hardest to understand, so I'm guessing I should ask: if sample paths were not continuous, what then?
As mentioned by the first comment, Brownian motion was developed with the goal to describe the "wriggling" movement of small particles (like pollen grains on the surface of water) due to the surrounding thermal bath, whose paths must be continuous, that is without jumps. It implies that the evolution of the system after an infinitesimal time step $\mathrm{d}t$ is itself infinitesimal, i.e. $\mathrm{d}B_t \sim \mathcal{N}(0,\mathrm{d}t)$. On the contrary, discontinuous paths have to be understood as jumpy processes; in other words, they can evolve through finite steps, even after an infinitesimal duration $\mathrm{d}t$. An example of such a process is given by the so called Poisson process.