Limits of applicability

A question: In physics, the world is considered to be continues. We can see this for example in the fact that we use integrals instead of finite sums in mechanics. However, we know that the objects in our world are made of atoms, which are in turn made of more elementary particles so they cannot be considered continues in a mathematical meaning. So can we apply mathematical result that are based on the assumption that the world is continues, to solve physical problems?

Since as I already said we use integrals in mechanics, it should be obvious that we can do this. But this is not true in all cases. For example lets consider the following theorem:

For any compact and convex set K in R^n and any continues function F from K to K, there is a point x in K such that F(x)=x.

This theorem is known as the general version of the Brouwer Fixed Point Theorem.

It should be obvious that a glass of water can be viewed as a compact and convex set in R^3. Also mixing the water using a spoon can be considered a continues function. Therefore, according to the theorem there is a point that didn't move. But this is obviously wrong. This is because the theorem gives us a point, but that point doesn't have to correspond to a particle and therefore saying that it didn't move is meaningless.

Why can we consider the world to be continues in one case and not the other? The reason for this is size. When we want to solve a problem in our scale (that is, we are dealing with objects that we can see and with results that can be observed directly), we can safely assume that the world is continues and use the corresponding tools. But this is only because the atoms are so small that for us there is no practical difference if we assume that the objects are continues.
But when like in the example with the water, the result we want to calculate (in that case a point) is not something we can see with our own eyes (we cannot see if an atom moved or not) assuming that the world is continues is wrong.

The above classification is very basic and inaccurate but it gives a general understating of the situation. In the next post, I plan to continue with this topic by discussing applicability of theories to reality.