Puntwise

Puntwise is a rarely used term in mathematics and computer science that denotes operations performed at each point of a domain, essentially behaving the same way as the more common term pointwise. The usage is sporadic, and many texts prefer “pointwise” to avoid confusion with other meanings of punt in different contexts.

Definition and notation

Let f and g be functions with the same domain X and codomain Y, and let a binary operation ⊗ on Y be given. A puntwise combination of f and g is the function h: X -> Y defined by h(x) = f(x) ⊗ g(x) for every x in X. The concept extends to unary operators: for a function φ: Y -> Z, the puntwise application is the function (φ puntwise f)(x) = φ(f(x)). In practice, f puntwise g describes a per-point operation where each input x yields a corresponding output determined by applying the chosen operation to the values f(x) and g(x).

Examples

Common examples mirror pointwise operations: (f puntwise g)(x) = f(x) + g(x) or (f puntwise g)(x) = f(x) · g(x). The notion also covers comparisons or transformations applied to each value, such as (f puntwise g)(x) = max(f(x), g(x)) or (φ puntwise f)(x) = φ(f(x)) for a fixed function φ.

Context and usage

In practice, puntwise is largely synonymous with pointwise, and many writers use the latter term exclusively. Puntwise appears occasionally in older or niche texts, or as a stylistic variation, and is generally understood in context to mean per-point evaluation. Related concepts include elementwise and per-element operations, which are common in programming and numerical analysis.