Like as once roots were generalized for negative numbers, I succeeded to generalize limits for arbitrary discontinuous functions.

The formula of limit of discontinuous function is based on algebraic general topology, my generalization of general topology in an algebraic way.

The formula that defines limit of discontinuous function is surprisingly simple:

lim f = { ν ∘ f ∘ r | r ∈ G }.

(This formula can be enhanced in different ways to make it behave better algebraically, but the idea is this.) And yes, it is very good algebraically, for example y – y = 0 as if it were just a real number!

So we can for example, define derivative of an arbitrary real function. It opens a way to wholly new discontinuous calculus.

There is a problem however: It is not easy to put this generalized derivative into a differential equation, because the left and the right parts of the equation would belong to different sets and we could not simply equate them. I solved this problem, too. So now we have a definition of non-differentiable (and even infinite) solutions of a differential equation.

What will happen next? I don’t know, but maybe for example, we will discover what is the structure at the point of singularity in a black hole.

Important details: My limit is defined not only for real and complex functions, but for example for every function from a vector topological space to a topological space, and even for wider cases including limit of multivalued function and wider!