That's true. I should have pointed out that from the developer's point of view it appears as if the execution is terminated, in the sense that no state is carried across code executions. However, the issue of whether the code is cold booted or not is an implementation detail of a specific serverless platform.
Atlassian did a nice blog post[1] on running Docker containers using socket activation. In principle that makes for a serverless platform but for a one that's difficult to manage. [1] https://developer.atlassian.com/blog/2015/03/docker-systemd-...
If you can't maintain state inside the application, you need to maintain it elsewhere (DB, Redis, memcache), and that elsewhere has a cost associated with accessing it.
Consider session state in a web app. If you keep state in memory and scale horizontally, you need session affinity, so stateless inbound http connections hit the same server instance. That means a session-aware router, delicate and extremely expensive F5s and other "smart" routers. But if the app is stateless and session state is in external storage, then all you need for load balancing is a simple round robin. A cheap router, that's more reliable and requires less administration, for a fraction of the cost.
There are other ways stateful apps are expensive, but that's a big one. I'd rather pay $5k than $100k for a router.
AWS Lambda in particular will spin up a new container running their framework, which will then stay alive for a period of time before being terminated.
With socket activation (using xinitd or systemd) and a minimal framework, you could implement much the same on your own VMs.