Below is actually a statement in textbook. But I don't have a good intuition of it.

If we want a stochastic process $W_t$ to satisfy i). $s\neq t$ implies $W_s$ and $W_t$ are independent, ii). $\{W_t\}$ is stationary, iii). $E[W_t]=0$ for all t, then $W_t$ cannot have continuous paths.

I hope someone can point out the essence of this argument to me. Also, Is there a continuous process satisfying the first two requirements?

Thanks!