Do infinitesimals exist in nature? My question is related to what is the physical content of infinitesimals. In mathematics the infinitesimals as hyperreals (Robinson 1960) are introduced as numbers greater than 0 but smaller than any real number. Can a physical quantity in real nature increase with a such infinitesimal number?
This must be outside physics because it is impossible to measure (each measurement gives a real number - even a rational decimal number). In the end infinitesimal increase can not be proven.
 A: Numbers, by themselves, do not have any physical content whatsoever. Physics is about creating models that accurately predict the outcome of experiments. So the physical content is in the model, not the numbers used by the model.
That said, the most successful models that we have developed, the ones that produce the best experimental predictions, all of those models can be based on infinitesimals.
However, the same models can also be based on limits. So people who don’t like infinitesimals can use the same models but based on limits instead of infinitesimals. Infinitesimals and limits are mathematically equivalent for these models. It is the models that have the physical content and not the underlying numbers.
A: The existence of planck length and plank time seems to show ourselves that the world is discrete....
So i think we can answer "no" to your question...
Everything in nature appears to be quantized at fundamental level
A: As @Dale has correctly pointed out, infinitesimals are a mathematical concept. They are relevant to physics as much as the fields and other mathematical objects in physical theories can be considered continuous, and therefore described in terms of differential equations. Specifically:

*

*On a microscopic level not everything is continuous - at least not all the time. Still, infinitesimals have their mathematical role: e.g., discrete spectra can be calculated and analyzed using continuous Schrodinger equation.

*Many fields of physics, notably the theory of elasticity and the macroscopic electrodynamics, operate with a physically small volume - that is the quantities in these theories are understood as averages over volumes containing $N_A$ number of atoms/molecules - not truly infinitesimal, but which ate treated as infinitesimally small in mathematical equations.

*Coarse graining is the term specifically used to describe the type of procedures that introduce continuous description of microscopically discrete problems, notably in the theory of the critical phenomena.

