When a dual-engine aircraft (such as the B737, A320, A330, B777 or B787) has reached the point where it is too late to abort a take off, can it take off if one engine fails?
It is a regulatory requirement (EASA CS 25) that commercial aircraft can safely fly should one of the two engines fail at any point of the flight, including takeoff at speeds beyond $V1$. (the maximum speed at which it can still stop before the end of the runway).
Control Following Engine Failure 1 An acceptable means of showing compliance with CS 25.143(b)(1) is to demonstrate that it is possible to regain full control of the aeroplane without attaining a dangerous flight condition in the event of a sudden and complete failure of the critical engine in the following conditions:
a. At each take-off flap setting at the lowest speed recommended for initial steady climb with all engines operating after take-off, with – i. All engines, prior to the critical engine becoming inoperative, at maximum take-off power or thrust; ii. All propeller controls in the take-off position; iii. The landing gear retracted; iv. The aeroplane in trim in the prescribed initial conditions; and
b. With wing-flaps retracted at a speed of 1.23 VSR1 with – i. All engines, prior to the critical engine becoming inoperative, at maximum continuous power or thrust; ii. All propeller controls in the en-route position; iii. The landing gear retracted; iv. The aeroplane in trim in the prescribed initial conditions.
For a clip of this happening just as the aircraft is rotating, take a look at this video:
A simulator video of the procedure is also available: