I have the following definition:
Definition (Poisson process). $N(t)$ is a poisson process if $N(t+h)-N(t)$ and $N(t)-N(t-k)$ are independent and $N(t+h)-N(t)\sim N(h)-N(0)$.
We try to find the law of such a process. In my course it's written that since $\mathbb P\{N(t)-N(t)=1\}=0$, we have that $$\mathbb P\{N(t+\Delta t)-N(t)=1\}=0+\lambda\Delta t+o(\Delta t).$$
I know it's the first order of taylor serie, but I don't understand how we can get it. What is the function ? (I guess it's $f(x)=\mathbb P\{N(t+x)-N(t)=1\}$, but what would be the derivative of such a function ?)
In the same way, how do we get $$\mathbb P\{N(t+\Delta t)-N(t)\geq 2\}=o(\Delta t)$$ and $$\mathbb P\{N(t+\Delta t)-N(t)=0\}=1-\lambda\Delta t+o(\Delta t)\ ?$$