Poisson and gaussian (or normal) distribution

[sanctus]

Just a little question: is it completely wrong to see the normal distribution as the generalization of the the Poisson distribution from only positive integers to the whole real axis?

[Pyrotex]

No, I don't think you have made a mistake.
In probabalistic risk analysis, we use the Normal as a generalization of the Poisson all the time. The Normal can be integrated, differentiated, making it more flexible and useful.