sum of continuous stochastic process and Gaussian process

Question

Suppose $A(t)$ is a continuous stochastic process and $B(t)$ is $N(0,1)$ for all $t$, then is the process $U$ \begin{equation*} U(t) = A(t) + B(t) \end{equation*} continuous or potentially discontinuous at some $t$?

I think this is a very stupid question, but I am somehow getting confused...

Answer 1

A counter example:

$$B(t) = \cases{X & $t\in \mathbb{Q}$ \\ -X & $t\in\mathbb{R} \setminus\mathbb{Q}$ }$$ for $X$ follows the normal distribution $\mathcal{N}(0,1)$

Then it's not difficult to verify that the process $U(t)$ is not continuous according to the definitions of continuity in this link.