A fundamental postulate of continuum fluid mechanics is that fluid motion can be described by a continuous, time-dependent mapping from a reference state to the current state. In other words, the paths of fluid particles can be described by a continuous function.
Addressing the topic of continuous fluid motion, Meyer (1971, pp. 5-6) describes a thought experiment on attempting to cut water with a knife. My version of Meyer's diagram is shown below. The configuration consists of a container of water, with air above, and a layer of dye in the water. A knife is passed through the water in an attempt to cut the line of dye. Rather than breaking, the layer of dye is indefinitely stretched by the moving knife. This is a nice illustration of continuous deformation of a fluid.
The no-slip condition in viscous fluid mechanics requires that a solid and the immediately adjacent fluid move at the same velocity. The same idea of continuous velocity holds where two fluids meet. A material surface in continuum fluid mechanics is a two-dimensional surface that always consists of the same particles and moves with the flow. Because of the no-slip condition, a fluid-solid interface in viscous flow is a material surface and a fluid-fluid interface is also a material surface (as long as diffusion across the interface is absent). In Meyer's thought experiment, the surface of the knife, the air-water interface, and the layer marked by dye are material surfaces. Because of the assumption of the continuous motion in fluid mechanics, material surfaces deform, but never break or disappear.
There are a couple of complications about the thought experiment worth considering. First, the knife is initially in air. Because of the no-slip condition, a layer of air must be present at the surface of the knife. Where does the air go as the knife enters the water? Does a very thin layer of air dissolve in the water? Second, the water closes up behind the knife as it enters. A section of the air-water interface disappears as the regions of water come together.
A slight variation of this experiment is shown in the video on the left below. Here, a tall glass of water has a layer of olive oil about 10 mm thick floating on top. I pass a knife through the olive oil into the water. The results support Meyer's thesis: the oil-water interface (a material surface) is greatly stretched, but does not break. Oil is seen to remain on the knife, as required by the no-slip condition.
Continuum fluid mechanics is not a perfect model of the physical world, and exceptions are known. Liquid interfaces can fracture or merge, processes that involve singularities under a continuum model of a free surface. Examples of singular behavior include the phenomena of breakup and coalescence of droplets and bubbles, and of rupture of films or sheets of liquid.
In the video to the right above, I try to illustrate these exceptions by redoing the experiment with the knife coated with dish detergent. (The blue bottle of detergent can be seen at the left in the video. Detergent is a surfactant; it reduces the surface tension at the oil-water interface.) Breakup of the layer of oil can be seen, especially in the later parts of the video. Perhaps some water has mixed with the soap at this stage. An ideal continuum model, with no diffusion and continuous material surfaces, is not consistent with these results.
I hope that you find this entertaining and stimulating of thought.
Reference: R. E. Meyer, Introduction to Mathematical Fluid Dynamics, J. Wiley, New York, 1971. Republished by Dover, 1982.