Kronecker delta instead of dirac delta as correlation function of white noise

From my understanding, if you have a sample $$x_{t_1},\dots,x_{t_n}$$ of $$X_{t_1},\dots,X_{t_n}$$ which are iid $$N(0,1)$$, then $$x_{t_1},\dots,x_{t_n}$$ is a sample path of Gaussian white noise.

However, it is stated that the correlation function $$c(s,t)$$ of white noise is $$\delta(s-t)$$ where $$\delta$$ is the Dirac delta function. I understand that if $$s\neq t$$ then $$c(s,t) = 0$$. However, if $$s=t$$, you want $$c(s-t)=1$$. Why isn't the kronecker delta function used instead?

• What is your definition of the Kronecker delta function? Be sure to identify all the symbols in what you write and tell us which are real numbers, which are integers, etc, Meanwhile, take a look at this answer. – Dilip Sarwate Mar 26 '20 at 3:31

• But still, let's say that $X(t)$ is Gaussian white noise so that it is a continuous time stochastic process. If $c(s,t) = \delta(s-t)$, is the correlation function, does that mean that if $s=t$ then $c(s,t) \neq 1$ since $\delta(s-t)$ is the Dirac delta function? – user1237300 Mar 26 '20 at 1:16