Equation of continuity says us that if we insert some fluid in a tube, the same amount of fluid will come out from the other end. If we make a small hole in a hose pipe, water will come out with a great speed. The bigger the hole, the slower the speed. This is a direct consequence of the equation of continuity.
But at the case of water tap, when we start to turn on the tap slightly, the velocity of water is slow. As we turn on the tap more, the speed increases. This is contradictory with equation of continuity.