Show that a stochastic process $X(t)$ is mean square continuous if and only if its autocorrelation function $R_X(t_1,t_2)$ is continous
$\Rightarrow$ Proof:
We have $E[(X(t)-X(t_0))^2]=R_X(t,t)-R_X(t_0,t)-R_X(t,t_0)+R_X(t_0,t_0)$, so if $t \rightarrow t_0$ the right side of the equation is equal to $0$, therefore $\Rightarrow$ implication is true.
$\Leftarrow$ Proof:
This is where I have trouble.
Is this implication true due to the fact that if $X(t)$ is continuous at $t_0$ in m.s then $\lim_{t \to t_0} m_X(t)=m_X(t_0)$ ??