# 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 $0$?

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 $W_t = (X_t+1)U$.