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The continuum hypothesis in continuum mechanics, and especially hydrodynamics, is the assumption that fluids can be modelled with functions of real numbers. Therefore, physical properties of fluids are described by time dependent scalar or vector fields on $\mathbb{R}$.
This assumption 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)
one quick comment: when I first saw the link, I thought it’s about a statement in set theory, not continuum mechanics.