quantitative - Black Scholes differential

I'm studying a BS derivation and I don't understand one part .We have a portfolio consisting of $\Delta(t)S(t)+B(t)$ where the first term is risky and the second is a riskless bond. The part i don't understand is why when we take the differential of this portfolio we obtain: $\Delta dS +dB$. I understand the reason is related to the fact that this is the limit of a discrete model, so we can imagine $\Delta$ as a function of $t$ with more steps of lenght $\Delta t$ so in the limit $\Delta t \to0$. But this would mean that $\frac{\partial \Delta}{\partial S}=0$ that is $\frac{\partial^2 O}{\partial S^2}=0$ where $O$ is the derivative. Could anyone explain me this passage?