fepsilonx

fepsilonx is a symbolic designation used in mathematical discussions to refer to a parametric family of functions f_epsilon(x) that depend on a small positive parameter epsilon. The term is not a standard object with a fixed definition; rather, it appears in expository texts and informal research to illustrate how a function can vary with a perturbation parameter and how its behavior changes as epsilon tends to zero.

Definition and scope: For a given domain X and codomain Y, fepsilonx denotes a family of maps f_epsilon: X -> Y indexed by epsilon > 0. In typical analyses, one studies the limit as epsilon approaches zero, f_0(x), and asks whether f_epsilon converges to f_0 uniformly on subsets of X, or pointwise. Convergence properties depend on the specific form of f_epsilon and the context of the problem.

Common instantiations: Examples include f_epsilon(x) = x/(1 + epsilon), which converges to x as epsilon -> 0, and f_epsilon(x) = x exp(-epsilon g(x)) which also tends to x for fixed x if g is bounded. Other forms may model regularization or smoothing and may converge to different limit functions depending on the design of f_epsilon.

Usage and context: In numerical analysis, fepsilonx is used to discuss discretization error, stabilization terms, or regularization parameters. In asymptotic analysis, it serves as a generic label for perturbation expansions where f_epsilon(x) = f_0(x) + epsilon f_1(x) + ...

Notational status: fepsilonx is not standardized notation; it is primarily a didactic or hypothetical device to illustrate perturbation ideas.