C172 Flight Instructor reported taking evasive action to avoid reported helicopter traffic resulted in an airspace violation and NMAC.

Narrative

I was departing ZZZ Airport when ATC advised there was a helicopter a few hundred feet below and about a mile off my left wind and I reported traffic in sights. ATC then contacted the helicopter off my right wing and informed him that I was off his right side and he acknowledged and reported he had me in sight. Shortly after that the helicopter began a climbing right turn towards my direction. The helicopter went out of my field of view due to me being in the right seat; I asked my student if he had visual contact with the helicopter but he did not give me a clear answer and started looking around frantically. My ADS-B was showing me the aircraft was directly below me and at an extremely close distance. I then decided to take control of the aircraft from my student and perform evasive maneuvers to avoid collision. I applied full power and began a steep climb before lowering my left wing to establish visual contact with the helicopter. I saw the aircraft directly below me at a very close range still climbing alongside me. I continued my climbing turn away from the helicopter and once I was clear I checked my altimeter and I was at 3;100 ft.; 100 ft. into the overlying class Bravo airspace beginning at 3;000. I quickly reduced throttle and returned to 2;800 ft. I believe without the evasive maneuvers I performed it could have resulted in collision. I am not aware of the helicopters tail number but it happened approximately 4nm south east of ZZZ airport on the edge of ZZZ airspace. I believe due to the high traffic area the helicopter may have not seen us and referred to another Cessna resulting in him beginning his turning climb directly into our path. I wouldn't imagine the pilot knowingly would have done that.