If a twin turbojet aircraft has an engine failure during takeoff, does it return for landing or does it continue to cruise with asymmetric thrust?
As a practical matter, the aircraft will probably land ASAP. Having lost an engine pretty clearly qualifies as an emergency for twin-engine aircraft!
That said, the aircraft would be capable of cruising with asymetric thrust. Rudder is applied to keep the aircraft flying straight.
Aircraft are allowed to takeoff in weather conditions that are below landing limits.
So yes, with an engine failure after takeoff, twin engine aircraft may plan a "Takeoff Alternate" up to one hour flight time away.
In cruise, modern twin jet airliners like the B777 and A330 are designed to fly ETOPS ("Extended-range Twin-engine Operational Performance Standards") for up to 330 minutes engine to a suitable landing field on ONE engine.
The jet will take off, but declare an urgent situation and either land as soon as practical if the aircraft is below gross landing weight or be directed to a holding area to either dump off or burn off fuel until the a/c is below gross landing weight, and then land as soon as practical.