When concepts (properties, operations, etc.) on a set $Y$ are extended to functions $f\colon X\longrightarrow Y$ by treating each function value $f(x)$ in isolation, the extended concept is often qualified with the word pointwise. One example is pointwise convergence of functions—a sequence $\{f_{n}\}_{{n=1}}^{\infty}$ of functions $X\longrightarrow Y$ converges pointwise to a function $f$ if $\lim_{{n\rightarrow\infty}}f_{n}(x)=f(x)$ for all $x\in X$.

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

where the identities hold for all $x\in X$. Pointwise operations inherit such properties as associativity, commutativity, and distributivity from corresponding operations on $Y$.

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