Continuity mistake: When Supergirl visits the principal's office, he is holding the glass down, close to the medicine bottle. When the angle changes, the glass is up without him having raised it.
Continuity mistake: When Supergirl wakes up in the woods, the baseball's stitches face different sides depending on the angle.
Continuity mistake: Selena's friend places the bottle of milk on the center of the burning chicken to extinguish it. When the angle changes the bottle is on the left side of the chicken.
Revealing mistake: When the invisible monsters throws Supergirl against the street light, the place she falls looks uneven with the rest of the lawn, and is filled with chunks of grass to hide the landing mat. In a wider shot all this atrezzo disappears.
Continuity mistake: Inside the girl's bathroom, when the red haired girl comes running in with her friend, watch the towel in the back: In one shot it's laying messy on the sink, and a second later it's neatly folded.
Deliberate mistake: For dramatic purposes, when the film ends, Supergirl leaves a cloudy trace while she flies. No trace was seen during her previous flights.
Continuity mistake: Supergirl arrives on earth, lands in a forest in broad daylight, walks four steps and when she stares at a lake it's suddenly dusk light.
Revealing mistake: Before Kara breaks the water container to fight the fire in the tires, the hole in the tank is already on it.
Continuity mistake: While Supergirl is flying around the city her bracelet is on her right hand then on her left.
Continuity mistake: Before Kara used the sphere to travel to earth she did not have the blue/red suit with the cape but suddenly the suit comes out from nowhere.
Continuity mistake: While Supergirl is talking with Zalter in Kripton beside them there are a couple with a man and a girl who disappear between shots and then reappear again.
Continuity mistake: When Supergirl flies over the field with the horses galloping below, the horses disappear in a couple of wide shots of her.
Revealing mistake: When Kara comes out from the lake, the wire cable that pulls her up is visible.
Continuity mistake: When Selena toys with the Omegahedron and it goes berzerk, she and Bianca hug. The position of her arms isn't continuous in any of the shots.
Continuity mistake: When Selena's car arrives at the campus she is not in the car. A shot later she appears inside.
Continuity mistake: Inside Selena's car, the first shot of the Omegahedron shows her thighs and the book on her lap. In the following shot Selena and the book vanish.
Continuity mistake: When Selena plays with her Tarot cards, the one on the far left corner has a round square drawing on the base. In the close-up it's a different card.
Continuity mistake: After the hockey ball crashes against Supergirl's back, a girl with glasses staring at her is holding her hockey club. A shot later her arm is raised.
Continuity mistake: When the school bus arrives at the campus there are only 3 or 4 girls inside. From the shots inside Selena's car at least 9 girls leave the bus.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.