Regulariteetti

Regulariteetti (regularity) is a broad concept used in Finnish to denote the property of being regular, orderly, or predictable. In mathematics it is often used to describe the smoothness or well-behavedness of objects such as functions, sets, or spaces. For a real-valued function, regularity indicates levels of smoothness: continuous (C0), differentiable (Ck), or smooth (C∞), with weaker notions such as Lipschitz or Hölder continuity also common. Regularity conditions are central to analysis because higher regularity often yields stronger conclusions about a solution’s behavior.

In partial differential equations, regularity theory studies when solutions inherit smoothness from equations and data. Elliptic regularity, for example, asserts that if coefficients and forcing terms are smooth, the solutions are smooth on the domain. Sobolev spaces provide a framework to quantify regularity in an integral sense.

In algebraic geometry, a point on a variety is regular (non-singular) if the local ring has dimension equal to the dimension of the variety; such regularity implies stable local geometry.

Outside pure mathematics, the term appears in probability (regularity of distributions and conditional probabilities), in physics (regularization as a method to control infinities in models), and in linguistics (regular versus irregular morphological patterns).

The term stems from Latin regularis, via French and other languages. In Finnish scholarly use, Regulariteetti captures the idea of orderliness, smoothness, and well-defined behavior across disciplines.