In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.
Calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless.
Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Jerome Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."
Read more about Non-standard Calculus: Motivation, Definition of Derivative, Continuity, Uniform Continuity, Compactness, Heine–Cantor Theorem, Why Is The Squaring Function Not Uniformly Continuous?, Example: Dirichlet Function, Limit, Limit of Sequence, Extreme Value Theorem, Intermediate Value Theorem, Basic Theorems, Applications
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