Consider the function f(x) = 0 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semicontinuous at x_{0} = 0, but not lower semicontinuous.
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 semicontinuous function which in general is not upper semicontinuous (for instance if you focus on the edge of a table).
Suppose X is a topological space, x_{0} is a point in X and f : X > R is a realvalued function. We say that f is upper semicontinuous at x_{0} if for every ε > 0 there exists a neighborhood U of x_{0} such that f(x) < f(x_{0}) + ε for all x in U. Equivalently, this can be expressed as
We say that f is lower semicontinuous at x_{0} if for every ε > 0 there exists a neighborhood U of x_{0} such that f(x) > f(x_{0})  ε for all x in U. Equivalently, this can be expressed as
A function is continuous at x_{0} if and only if it is upper and lower semicontinuous there.
If f and g are two functions which are both upper semicontinuous at x_{0}, then so is f + g. If both functions are nonnegative, then the product function fg will also be upper semicontinuous at x_{0}. Multiplying a positive upper semicontinuous function with a negative number turns it into a lower semicontinuous 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 semicontinuous functions and minima is also true.
Suppose f_{n} : X > R is a lower semicontinuous function for every natural number n, and
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