I have read the definitions of Additive White Gaussian Noises as,
$S_x(f) = \frac{N_0}{2}$
This gives me (after an inverse Fourier Transform),
$R_x(\tau) = \frac{N_0 \delta(\tau)}{2}$
Now, $R_x(0) = E(X^2(t)) = \sigma^2$ (the variance of the random variable, a finite quantity)
However, on putting $\tau = 0$ in $R_x(\tau)$, I obtain an undefined value. What is the flaw in this argument?