I came across the following lines that appear after the derivation of equation of continuity for the steady flow of an ideal liquid in Resnick, Halliday, Kranes's Fundamentals of Physics:
The equation of continuity states that if within any volume element of space (not volume of fluid) there are no sources(where additional matter is introduced into the flow) or sinks(where matter is removed from the flow),then the total mass within the volume element must remain constant. In more general cases, if sources or sinks are present, the equation of continuity gives the mathematical representation of the very reasonable assertion that the rate of outflow OR inflow of matter is equal to the rate at which the mass contained in the volume element is changing.
Now, the statement in bold is what is troubling me. Is it not true that the mass contained in a volume element is simply ρdV where ρ is the density of the liquid at the location of some volume element of size dV. Now, since steady flow is assumed, ρ does not change with time. Hence, regardless of whether there are sources or sinks in the element, the mass of the fluid within the element also should not change with time ( as the mass is a function of only ρ and dV which is also constant). So where comes the question of "mass contained in the volume element changing"(refer blockquote)?
Also note the usage of OR in the passage. Does it mean the the rate of outflow is equal to rate of inflow? Are these two rates equal even when there are sources or sinks in the element?
Can someone please elaborate?