Lower envelope

The concept of a lower envelope can also be extended to partial functions by taking the minimum only among functions that have values at the point.

The upper envelope or pointwise maximum is defined symmetrically.

For an infinite set of functions, the same notions may be defined using the infimum in place of the minimum, and the supremum in place of the maximum.

For functions of a single real variable whose graphs have a bounded number of intersection points, the complexity of the lower or upper envelope can be bounded using Davenport–Schinzel sequences, and these envelopes can be computed efficiently by a divide-and-conquer algorithm that computes and then merges the envelopes of subsets of the functions.

However, the lower and upper envelope operations do not necessarily preserve the property of being a continuous function.