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The continuum hypothesis in continuum mechanics, and especially hydrodynamics, is the assumption that fluids can be approximately modelled with functions of real numbers. (There is also a statement in set theory called the ‘continuum hypothesis’, but this is completely unrelated.)
The continuum hypothesis justifies our description of fluids by time dependent scalar or vector fields on $\mathbb{R}$. This can only be approximately true, of course, since the most accurate description of matter known today uses the concept of elementary particles and atoms of particles.
For liquids, the continuum hypothesis is a good approximation for most practical situations, for gases, the Knudsen number K should be much smaller than one, $K \ll 1$, where K is defined to be $K:= \frac{\lambda}{L}$. $\lambda$ is the mean free path of a particle, and $L$ is the length scale of phenomena that one wishes to describe.
The continuum hypothesis, Wikipedia.
Pieter Wesseling: Computational Fluid Dynamics (ZMATH)
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