Let $N=(N(t))_{t\geq 0}$ be a Poisson process with continuous intensity function $(\lambda(t))_{t\geq0} $.
a) Show that the intensities $\lambda_{n, n+k}(t)$, $n\geq0$, $k\geq1$ and $t>0$, of the Matov process $N$ with transition probabilities $p_{n,n+k}(s,t)$ exist, i.e., $$\lambda_{n, n+k}(t)=\lim_{h\downarrow0}\frac{p_{n,n+k}(t,t+h)}{h}, \text{ } n\geq0 \text{ }, k\geq1,$$ and that they are given by $$\lambda_{n,n+k}(t)= \begin{cases} \lambda(t), k=1 \\ 0, k\geq2\end{cases} .$$
b) What can you conclude from $p_{n,n+k}(t,t+h)$ for $h$ small about the short term jump behavior of the Markov process $N$?
c) Show by counterexample that system from a) in is general not valid if one gives up the assumption of continuity of the intensity function $\lambda(t)$.
So I started to learn about Poisson process by myself. But this looks really hard to me... didn't find any examples how should I solve exercises like this one. Anybody have any ideas?