Well, I missed this earlier.
For those that don't know, in Italy you don't vote for the Prime Minister. You vote for the parliament members, then the president of the republic has to call someone (almost always the secretary of the biggest party in the coalition that "won") and ask him to find the majority in Camera and Senato. If said person and his government successfully gets the "fiducia" in both houses, then it becomes officially the italian government. This is not a given, recent example is Bersani in 2013 that didn't want to have an alliance with Berlusconi and asked M5S for a while, but they kept refusing (they would've lost "votes", they were the pure and honest ones) so Napolitano had to ask Letta to form one (which he did with the right).
When a PM resigns you're not forced to go to elections because of that. Parliament members are supposed to stay for the whole term (called "legislatura"), so the PoR can just ask someone else to form a government (happened with Monti in 2011, which had the majority thanks to Berlusconi's party). But even if the legislatura is not over, the PoR can call elections early.
This system is so dumb that Renzi I was the 4th longest serving government since WW2 and the 8th "ever" (since 1861).
source in italian
Can anyone think of a more unstable electoral law? Because I can't.