Another formula for second partial derivative of covariance function

Question

So the problem arises from Stochastic Processes theory (particularly covariance function of the derivative of $L_2$-process) and is formulated like this:

It can be proved that

$$C_{X'}(s,t)=\lim_{(\xi,\tau)\to(s,t)}\frac{C(\xi,\tau)-C(\xi,t)-C(s,\tau)+C(s,t)}{(\xi-s)(\tau-t)} $$ Where $C$ - covariance function of process $X\quad (C(s,t)\stackrel{def}{=}\mathbb{E}(X(s)-\mathbb{E}X(s))(X(t)-\mathbb{E}X(t)))$ ;
$C_{X'}$ - covariance function of process $X'(t)\quad (\lim_{\tau\to t}\mathbb{E}\left|X'(t)-\frac{X(\tau)-X(t)}{\tau-t}\right|^2\stackrel{def}{=}0)$;
Since this limits is very similar to $\frac{\partial^2{C}}{\partial{s}\partial{t}}(s,t)$.
What are the sufficient conditions, function $f$ must satisfy for equality to hold
i.e. in what case? $$\lim_{(\xi,\tau)\to(s,t)}\frac{C(\xi,\tau)-C(\xi,t)-C(s,\tau)+C(s,t)}{(\xi-s)(\tau-t)}=\frac{\partial^2{C}}{\partial{s}\partial{t}}(s,t) $$

I think it's worth mentioning that

$$\lim_{(\xi,\tau)\to(s,t)}f(\xi,\tau)\neq\lim_{\xi\to s}\lim_{\tau\to t}f(\xi,\tau)$$ in general case.

I will be very grateful for any useful information like books, where such problem potentially may be be solved and also for your own thoughts or even hints on this subject. Thanks for your time.