Semi-continuous

A real-valued function f is upper semi-continuous at a point x₀ if, roughly speaking, the function values for arguments near x₀ do not rapidly jump upwards. If they don't rapidly jump downwards, the function is called lower semi-continuous at x₀.

Examples

Consider the function f(x) = 0 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semi-continuous at x₀ = 0, but not lower semi-continuous.

Imagine that you are scanning a certain scenery with your eyes and record the distance to the viewed object at all times. This yields a lower semi-continuous function which in general is not upper semi-continuous (for instance if you focus on the edge of a table).

Formal definition

Suppose X is a topological space, x₀ is a point in X and f : X -> R is a real-valued function. We say that f is upper semi-continuous at x₀ if for every ε > 0 there exists a neighborhood U of x₀ such that f(x) < f(x₀) + ε for all x in U. Equivalently, this can be expressed as

lim sup_x→x0 f(x) ≤ f(x₀).

The function f is called upper semi-continuous if it is upper semi-continuous at every point of its domain.

We say that f is lower semi-continuous at x₀ if for every ε > 0 there exists a neighborhood U of x₀ such that f(x) > f(x₀) - ε for all x in U. Equivalently, this can be expressed as

lim inf_x→x0 f(x) ≥ f(x₀).

The function f is called lower semi-continuous if it is lower semi-continuous at every point of its domain.

Properties

A function is continuous at x₀ if and only if it is upper and lower semi-continuous there.

If f and g are two 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 interval [a, b]) and f : C -> R is upper semi-, then f has a maximum on C. The analogous statement for lower semi-continuous functions and minima is also true.

Suppose f_n : X -> R is a lower semi-continuous function for every natural number n, and

f(x) := sup {f_n(x) : n in N} < ∞

for every x in X. Then f is lower semi-continuous. Even if all the f_n are continuous, f need not be continuous.