Final answer:
The question pertains to real analysis and the existence of an increasing function that is discontinuous precisely at irrational numbers. While no classical function directly fits this description, real analysis offers methods to explore and potentially construct such functions.
Step-by-step explanation:
The question involves a concept in the field of real analysis, which is a branch of mathematics. Specifically, it asks whether there exists an increasing function whose points of discontinuity coincide exactly with the set of irrational numbers. To provide some insight, we can discuss properties of functions and their discontinuities.
However, the actual existence of such a function is a more complex matter. There is a well-known construction in real analysis known as the Thomae's function, which is discontinuous at every rational point and continuous at every irrational point.
While this function doesn't directly answer the question, since it's not strictly increasing, it highlights the intricate relationship between continuity and the nature of the domain.
Answering this question would require in-depth exploration of function properties and potentially the construction of a novel function or the adaptation of known functions to meet the criteria.