:for the notion of upper or lower semicontinuous multivalued function see: hemicontinuity in mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. an extended real-valued function f is upper (respectively, lower) semi-continuous at a point x0 if, roughly speaking, the function values for arguments near x0 are either close to f(x0) or less than (respectively, greater than) f(x0). == examples == thumb|right|an upper semi-continuous function. the solid blue dot indicates f(x0). consider the function f, piecewise defined by f(x)&nbsp;= â1 for x&nbsp;<&nbsp;0 and f(x)&nbsp;=&nbsp;1 for x&nbsp;â¥&nbsp;0. this function is upper semi-continuous at x0&nbsp;=&nbsp;0, but not lower semi-continuous. thumb|right|a lower semi-continuous function. the solid blue dot indicates f(x0). 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 f(x)=\lfloor x \rfloor, which returns the greatest integer less than or equal to a given real number x, is everywhere upper semi-continuous. similarly, the ceiling function f(x)=\lceil x \rceil is lower semi-continuous. a function may be upper or lower semi-continuous without being either left or right continuous. for example, the function :f(x) = \begin{cases} 1 & \mbox{if } x 1, \end{cases} 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 : f(x) = \begin{cases} \sin(1/x) & \mbox{if } x \neq 0,\\ 1 & \mbox{if } x = 0, \end{cases} is upper semi-continuous at x = 0 while the function limits from the left or right at zero do not even exist. if x=\mathbb r^n is a euclidean space (or more generally, a metric space) and \gamma=c([0,1],x) is the space of curves in x (with the supremum distance d_\gamma(\alpha,\beta)=\sup_t\ d_x(\alpha(t),\beta(t)), then the length functional l:\gamma\to[0,+\infty], which assigns to each curve \alpha its length l(\alpha), is lower semicontinuous. let (x,\mu) be a measure space and let l^+(x,\mu) denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to \mu. then the integral, seen as an operator from l^+(x,\mu) to [-\infty,+\infty] is lower semi-continuous. this is just fatou's lemma. == formal definition == suppose x is a [[topolo
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