Prove that a stochastic process cannot have continuous paths

Question

Consider the following problem.

Suppose that a stochastic process $V$ satisfies the conditions below

• $t_1 \neq t_2$ implies that $V(t_1)$ and $V(t_2)$ are independent.
• $\{V(t)\}$ is stationary, i.e. the (joint) distribution of $\{V(t_1+t),\dots,V(t_k+t)\}$ does not depend on $t$.
• $\mathbb{E}[V(t)] = 0$ for all $t$. 

Prove that $V(t)$ cannot have continuous paths.

The hint says to consider $E[|V(t,N)-V(s,N)|^2]$ where $$ V(t,N) = \max\{-N,\min\{N,V(t)\}\}, \quad N=1,2,3,\dots $$ I don't see how this hint helps to prove that $V$ cannot have continuous paths. The only connection I found is the Kolmogorov's continuity Theorem.

Any help is appreciated!!