Examples
Consider the function f, piecewise defined by f(x) = –1 for x < 0 and f(x) = 1 for x ≥ 0. This function is upper semi-continuous at x0 = 0, but not lower semi-continuous.
The indicator function of an open set is lower semi-continuous, whereas the indicator function of a closed set is upper semi-continuous. The floor function, which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. Similarly, the ceiling function is lower semi-continuous.
A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function
is upper semi-continuous at x = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function
is upper semi-continuous at x = 0 while the function limits from the left or right at zero do not even exist.
Let be a measure space and let denote the set of positive measurable functions endowed with the topology of -almost everywhere convergence. Then the integral, seen as an operator from to is lower semi-continuous. This is just Fatou's lemma.
Read more about this topic: Semi-continuity
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