Show that $ X_t = \begin{cases} W_t & \text{if $t \neq a$ } \\ 0 & \text{if $t = a$ } \end{cases} $ is not a Brownian motion

Question

Let $W_t$ be a Brownian motion and $a \in (0,T)$. Consider the function $$ X_t = \begin{cases} W_t & \text{if $t \neq a$ } \\ 0 & \text{if $t = a$ } \end{cases} $$

Show that $X_t$ is not a Brownian motion.

Here is my work: I want to prove that the increments of $X_t$ are correlated. Let $s<a<t$, $Cov(X_a - X_s, X_t-X_a) = Cov(-W_s,W_t) = -s$. Hence, the increments are not independent and $X_t$ is not a Brownian motion.

I'm not sure if I prove it right as I just picked one special case.

Answer 1

We need $$ X_a \sim \mathcal{N}(0,a) $$ which is not the case.