Main Page | See live article | Alphabetical index In mathematical analysis, semicontinuity is a property of real-valued functions that is weaker than continuity. It comes in two kinds, upper semicontinuity and lower semicontinuity. A real-valued function over a topological space is said to be lower semicontinuous if the following property holds: is an open set for every . It is said to be an upper semicontinuous function if the following property holds: is an open set for every . Properties A function is continuous if and only if it is both upper and lower semicontinuous. The characteristic function of an open set is lower semicontinuous. The characteristic function of a closed set is upper semicontinuous. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.