The Continuity of Correlation Coefficient of a Continuous Stochastic Process Given a continuous stochastic process with respect to time with finite variance at a given $t$. Does it necessarily imply, as $d\to 0$,
1) the covariance between $t$ and $t+d$ approaches the variance at $t$?
2) when both variance at $t$ and $t+d$ are positive, the correlation coefficient between $t$ and $t+d$ approaches $1$?
3) if the answer to 2) is in the negative, that the correlation between $t$ and $t+d$ can not approach $0$?
 A: No to questions 1 and 2.
Let $X_t$ be the process defined in this answer, which is continuous, satisfies $E X_t = 0$ for $t < 1$, and for $t \ge 1$ we have $X_t = 1$ almost surely.  Let $U$ be any random variable with nonzero finite variance, and independent of the process $\{X_t\}$.  Set $V_t = U X_t$, which is also continuous.  Since $X_t$ and $U$ are nonconstant, $\operatorname{Var}(V_t) > 0$ for all $t$.  But for $t < 1$ we have
$$
\begin{align*}
\operatorname{Cov}(V_t, V_1) &= E[V_t V_1] - E[V_t] E[V_1] \\
&= E[X_t U^2] - E[X_t U] E[U] \\
&= E[X_t] E[U^2] - E[X_t] E[U]^2 && \text{by independence} \\
&= 0
\end{align*}
$$
since $E[X_t] = 0$.  This resolves your questions 1 and 2 in the negative.
For question 3, I don't have an example off the top of my head, but I would probably try a process like $W_t = (X_t+1)U$.
Generally, the continuity of a stochastic process is a statement about almost sure convergence, and you really should not expect almost sure convergence to imply anything about convergence of moments.
