What does the continuum hypothesis of fluid mechanics mean? I'm a bit confused by the continuum hypothesis stating that fluid are continuous objects rather than made out of discrete objects. 
Say for $\rho (x,t)$ (density) is there more than one fluid particle at $x$ or less than one.
 A: The continuum hypothesis means the following: at each point of the region of the fluid it is possible to construct one volume small enough compared to the region of the fluid and still big enough compared to the molecular mean free path.
Why is that important? Because of two things. First, since the volume you can build at each point is very small compared to the size of the region of the fluid, you can think of the volume as located at a point instead of considering it as a collection of points. Imagine Earth for instance. If you build a small volume of $1 m^3$ somewhere on the surface of the Earth it is so small compared to Earth's size it can be considered to be associated with a particular point.
The secont thing is that since the volume is big enough compared to the molecular mean free path, this means it contains a large enough number of molecules. Why is that something you would want? Because containing a reasonable number of molecules allows you to take means on the volume and those means will make sense.
So, for instance, you can go there, compute the mass of each molecule, sum them up and divide by the volume. Given this hypothesis, this mean makes sense. And given the first hypothesis, you can think about this mean as associated with the point.
Because of that it makes sense of talking about fields defined on the region of the fluid. The mass density for example or the velocity field. They are in truth, means of quantities associated with the molecules, but that on the macroscopic point of view, can be considered just as fields associating quantities to points of the region.
On the density case, it really means: if $x$ is a point on the region of the fluid and $t$ an instant of time, $\rho(x,t)$ is the mean value of the mass of molecules contained inside one such small volume associated at $x$ at time $t$. From the macroscopic point of view, it is just a density that allows you to get mass through integration.
