Apparent paradox in equation of continuity 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.
 A: The continuity equation says that IF VOLUMETRIC FLOW IS FIXED, e.g. "always XYZ liters per second", then a smaller pipe or smaller hole causes higher fluid velocity.
The water faucet is not like that. The flow is not fixed. When the faucet is slightly open it is "X liters per second" flowing out ... when the faucet is fully open it is "Y liters per second" flowing out. Y is bigger than X. It's NOT a constant fixed flow rate. Therefore the continuity equation does NOT imply that a narrower opening means higher fluid velocity.
A: If you have a tube of constant volume (like your water pipe), filled with an incompressible fluid (like water), then if any comes out, the same amount goes in.
So when your faucet is turned off, no water comes out, and no water goes into the other end of the pipe.
When you turn it on a little bit, a little water comes out, and the same amount goes in the other end.
If you turn it on a lot, same idea.
Don't let yourself be confused between the amount of water, and its speed.
A: The continuity equation is valid, however it doesn't say that the flow of water is the same in the two cases.
Let's see how to apply it correctly:


*

*The faucet is closed:
Water coming out of the aqueduct: $0\ m^3s^{-1}$
Water coming out of the faucet: $0\ m^3s^{-1}$

*The faucet is slightly open:
Water coming out of the aqueduct: $V_0\ m^3s^{-1}$
Water coming out of the faucet: $V_0\ m^3s^{-1}$

*The faucet is completely open:
Water coming out of the aqueduct: $V_1\ m^3s^{-1}$
Water coming out of the faucet: $V_1\ m^3s^{-1}$
So, in practice the equation is always valid, but it doesn't dictate that the water comes out of the faucet at a constant flow.
The law governing a faucet is Bernoulli's law which tells us that the pressure at the faucet will be constant, hence - since the density of the water is constant - the velocity of the water coming out, and thus the flow will be governed by the section of the flow let free by the faucet.
