While the answers given so far are true in current practice and certainly when switching between your mobile internet provider and home internet provider, both IPv4 and IPv6 do have mobility solutions, IPv4 by extension and IPv6 natively. They enable a mobile user to have a static home IP address and a dynamic mobile IP address which allows TCP connections to be maintained while moving from network to network. The details can be found in the RFCs below:
Mobile IP for IPv4 RFC
As far as the current deployment of these mobility solutions go, I am not aware of any, and I am not able to find anymore information. I would assume IPv4 mobility's solution is too cumbersome to deploy (it was an afterthought after all) and there aren't enough IPv4 addresses to go around as it is, let alone for mobile hence the heavy NAT utilization, so it won't ever be used. As for IPv6 and the reasons for its lack of widespread deployment is more difficult to speculate on, I would guess that with much of the web still on IPv4 and thus heavy use of IPv4 to IPv6 translation, the mobile solution could be tough to deploy in this environment. Regardless, even if a mobile provider did support the above, both your home and mobile provider would need some level of support, and home ISPs don't have any monetary incentive in this regard.
TL;DR TCP connections do break on your phone when changing networks (and thus network providers); however, there are mobility solutions that address this issue, but they haven't been deployed (at least not widely).
PS A better test of TCP connection breaking is a voice or video call (over data obviously). Music and movies can be buffered and the IP address transition is not noticed; however, calls cannot.