Semi-continuity - Properties

Properties

A function is continuous at x₀ if and only if it is upper and lower semi-continuous there. Therefore, semi-continuity can be used to prove continuity.

If f and g are two real-valued functions which are both upper semi-continuous at x₀, then so is f + g. If both functions are non-negative, then the product function fg will also be upper semi-continuous at x₀. Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function.

If C is a compact space (for instance a closed, bounded interval ) and f : C → -valued lower semi-continuous functions and minima is also true. (See the article on the extreme value theorem for a proof.)

Suppose f_i : X → is a lower semi-continuous function for every index i in a nonempty set I, and define f as pointwise supremum, i.e.,

Then f is lower semi-continuous. Even if all the f_i are continuous, f need not be continuous: indeed every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions.

The indicator function of any open set is lower semicontinuous. The indicator function of a closed set is upper semicontinuous. However, in convex analysis, the term "indicator function" often refers to the characteristic function, and the characteristic function of any closed set is lower semicontinuous, and the characteristic function of any open set is upper semicontinuous.

A function f : Rn→R is lower semicontinuous if and only if its epigraph (the set of points lying on or above its graph) is closed.

A function f : X→R, for some topological space X, is lower semicontinuous if and only if it is continuous with respect to the Scott topology on R.