Definition: $$ \frac{dx}{dt} = \eta(t), $$ where \(\eta(t)\) satisfies $$ \langle \eta(t)\eta(t')\rangle = \sigma^2 \delta(t-t'). $$ We want to know \(\langle dx^2\rangle\) over an infinitesimal time interval \(dt\). To calculate this, we further divide \(dt\) into \(N\) equal parts, where \(N\) is a large number. We have $$ dx = \left[\eta(dt/N) + \eta(2dt/N) + \cdots + \eta(dt)\right] \frac{dt}{N}. $$ We assume the \(\eta\)s in the bracket are independent and identically distributed. This gives $$ \langle dx^2\rangle = \left[\eta^2(dt/N) + \eta^2(2dt/N) + \cdots + \eta^2(dt)\right] \left(\frac{dt}{N}\right)^2 \\ = N \frac{\sigma^2}{dt/N} \left(\frac{dt}{N}\right)^2 = \sigma^2 dt. $$ Here we have used $$ \eta^2(dt/N) \approx \frac{\sigma^2} {dt/N}. $$ We could have assumed \(\eta^2 dt = \sigma^2\) from the beginning, but the mental image is a bit different. In \(\eta^2 dt\), we imagine \(dt\) is still somehow "large", although it is an infinitesimal, while for \(dt/N\), we assume it is an unsplittable "atomic" time interval, within which \(\eta\) is constant (not fluctuating). Namely, we were somehow choosing \(N\) such that \(dt/N\) equals exactly the "natural" "atomic" time. In reality, the "atomic" time may be the shortest time a system variable can change its value. This is reasonable, because we can't imagine something really happens within an infinitely small amount of time. Everything takes a nonzero amount of time to happen.

And we have seen that discretization makes things solid and clear.