Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value $f(x)$ of some function $f.$ An example is pointwise convergence of functions — a sequence of functions :$\{f\_n\}\_\{n=1\}^infty$with:$f\_n:X\; longrightarrow\; Y$
converges pointwise to a function $f$ if for each $x$ in $X$:$lim\_\{n\; ightarrow\; infty\}\; f\_n(x)\; =\; f(x).$

An important class of pointwise concepts are the "pointwise operations" — operations defined on functions by applying the operations to function values separately for each point in the domain of definition. These include

Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain.

An example of an operation on functions which is "not" pointwise is convolution.

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