Semi-continuity

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function ${\displaystyle f}$ is upper (respectively, lower) semicontinuous at a point ${\displaystyle x_{0}}$ if, roughly speaking, the function values for arguments near ${\displaystyle x_{0}}$ are not much higher (respectively, lower) than ${\displaystyle f\left(x_{0}\right).}$

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point ${\displaystyle x_{0}}$ to ${\displaystyle f\left(x_{0}\right)+c}$ for some ${\displaystyle c>0}$, then the result is upper semicontinuous; if we decrease its value to ${\displaystyle f\left(x_{0}\right)-c}$ then the result is lower semicontinuous.

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.^[1]

Definitions

Assume throughout that ${\displaystyle X}$ is a topological space and ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is a function with values in the extended real numbers ${\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]}$.

Upper semicontinuity

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called upper semicontinuous at a point ${\displaystyle x_{0}\in X}$ if for every real ${\displaystyle y>f\left(x_{0}\right)}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ such that ${\displaystyle f(x)<y}$ for all ${\displaystyle x\in U}$.^[2] Equivalently, ${\displaystyle f}$ is upper semicontinuous at ${\displaystyle x_{0}}$ if and only if

where lim sup is the limit superior of the function ${\displaystyle f}$ at the point ${\displaystyle x_{0}}$.

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called upper semicontinuous if it satisfies any of the following equivalent conditions:^[2]

(1) The function is upper semicontinuous at every point of its domain.

(2) All sets ${\displaystyle f^{-1}([-\infty ,y))=\{x\in X:f(x)<y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are open in ${\displaystyle X}$, where ${\displaystyle [-\infty ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}}$.

(3) All superlevel sets ${\displaystyle \{x\in X:f(x)\geq y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are closed in ${\displaystyle X}$.

(4) The hypograph ${\displaystyle \{(x,t)\in X\times \mathbb {R} :t\leq f(x)\}}$ is closed in ${\displaystyle X\times \mathbb {R} }$.

(5) The function is continuous when the codomain ${\displaystyle {\overline {\mathbb {R} }}}$ is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals ${\displaystyle [-\infty ,y)}$.

Lower semicontinuity

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called lower semicontinuous at a point ${\displaystyle x_{0}\in X}$ if for every real ${\displaystyle y<f\left(x_{0}\right)}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ such that ${\displaystyle f(x)>y}$ for all ${\displaystyle x\in U}$. Equivalently, ${\displaystyle f}$ is lower semicontinuous at ${\displaystyle x_{0}}$ if and only if

where ${\displaystyle \liminf }$ is the limit inferior of the function ${\displaystyle f}$ at point ${\displaystyle x_{0}}$.

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called lower semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is lower semicontinuous at every point of its domain.

(2) All sets ${\displaystyle f^{-1}((y,\infty ])=\{x\in X:f(x)>y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are open in ${\displaystyle X}$, where ${\displaystyle (y,\infty ]=\{t\in {\overline {\mathbb {R} }}:t>y\}}$.

(3) All sublevel sets ${\displaystyle \{x\in X:f(x)\leq y\}}$ with ${\displaystyle y\in \mathbb {R} }$ are closed in ${\displaystyle X}$.

(4) The epigraph ${\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}$ is closed in ${\displaystyle X\times \mathbb {R} }$.

(5) The function is continuous when the codomain ${\displaystyle {\overline {\mathbb {R} }}}$ is given the right order topology. This is just a restatement of condition (2) since the right order topology is generated by all the intervals ${\displaystyle (y,\infty ]}$.

Examples

Consider the function ${\displaystyle f,}$ piecewise defined by:

This function is upper semicontinuous at ${\displaystyle x_{0}=0,}$ but not lower semicontinuous.

The floor function ${\displaystyle f(x)=\lfloor x\rfloor ,}$ which returns the greatest integer less than or equal to a given real number ${\displaystyle x,}$ is everywhere upper semicontinuous. Similarly, the ceiling function ${\displaystyle f(x)=\lceil x\rceil }$ is lower semicontinuous.

Upper and lower semicontinuity bear no relation to continuity from the left or from the right for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.^[3] For example the function

is upper semicontinuous at ${\displaystyle x=0}$ while the function limits from the left or right at zero do not even exist.

If ${\displaystyle X=\mathbb {R} ^{n}}$ is a Euclidean space (or more generally, a metric space) and ${\displaystyle \Gamma =C([0,1],X)}$ is the space of curves in ${\displaystyle X}$ (with the supremum distance ${\displaystyle d_{\Gamma }(\alpha ,\beta )=\sup\{d_{X}(\alpha (t),\beta (t)):t\in [0,1]\}}$), then the length functional ${\displaystyle L:\Gamma \to [0,+\infty ],}$ which assigns to each curve ${\displaystyle \alpha }$ its length ${\displaystyle L(\alpha ),}$ is lower semicontinuous.^[4] As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length ${\displaystyle {\sqrt {2}}}$.

Let ${\displaystyle (X,\mu )}$ be a measure space and let ${\displaystyle L^{+}(X,\mu )}$ denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to ${\displaystyle \mu .}$ Then by Fatou's lemma the integral, seen as an operator from ${\displaystyle L^{+}(X,\mu )}$ to ${\displaystyle [-\infty ,+\infty ]}$ is lower semicontinuous.

Properties

Unless specified otherwise, all functions below are from a topological space ${\displaystyle X}$ to the extended real numbers ${\displaystyle {\overline {\mathbb {R} }}=[-\infty ,\infty ].}$ Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.

• A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is continuous if and only if it is both upper and lower semicontinuous.
• The indicator function of a set ${\displaystyle A\subset X}$ (defined by ${\displaystyle \mathbf {1} _{A}(x)=1}$ if ${\displaystyle x\in A}$ and ${\displaystyle 0}$ if ${\displaystyle x\notin A}$) is upper semicontinuous if and only if ${\displaystyle A}$ is a closed set. It is lower semicontinuous if and only if ${\displaystyle A}$ is an open set.^{[note 1]}

• The sum ${\displaystyle f+g}$ of two lower semicontinuous functions is lower semicontinuous^[5] (provided the sum is well-defined, i.e., ${\displaystyle f(x)+g(x)}$ is not the indeterminate form ${\displaystyle -\infty +\infty }$). The same holds for upper semicontinuous functions.
• If both functions are non-negative, the product function ${\displaystyle fg}$ of two lower semicontinuous functions is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
• A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is lower semicontinuous if and only if ${\displaystyle -f}$ is upper semicontinuous.
• The composition ${\displaystyle f\circ g}$ of upper semicontinuous functions is not necessarily upper semicontinuous, but if ${\displaystyle f}$ is also non-decreasing, then ${\displaystyle f\circ g}$ is upper semicontinuous.^[6]
• The minimum and the maximum of two lower semicontinuous functions are lower semicontinuous. In other words, the set of all lower semicontinuous functions from ${\displaystyle X}$ to ${\displaystyle {\overline {\mathbb {R} }}}$ (or to ${\displaystyle \mathbb {R} }$) forms a lattice. The same holds for upper semicontinuous functions.
• The (pointwise) supremum of an arbitrary family ${\displaystyle (f_{i})_{i\in I}}$ of lower semicontinuous functions ${\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}$ (defined by ${\displaystyle f(x)=\sup\{f_{i}(x):i\in I\}}$) is lower semicontinuous.^[7]

In particular, the limit of a monotone increasing sequence ${\displaystyle f_{1}\leq f_{2}\leq f_{3}\leq \cdots }$ of continuous functions is lower semicontinuous. (The Theorem of Baire below provides a partial converse.) The limit function will only be lower semicontinuous in general, not continuous. An example is given by the functions ${\displaystyle f_{n}(x)=1-(1-x)^{n}}$ defined for ${\displaystyle x\in [0,1]}$ for ${\displaystyle n=1,2,\ldots .}$

Likewise, the infimum of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a monotone decreasing sequence of continuous functions is upper semicontinuous.

• (Theorem of Baire)^{[note 2]} Assume ${\displaystyle X}$ is a metric space. Every lower semicontinuous function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is the limit of a monotone increasing sequence of extended real-valued continuous functions on ${\displaystyle X}$; if ${\displaystyle f}$ does not take the value ${\displaystyle -\infty }$, the continuous functions can be taken to be real-valued.^[8][9]

And every upper semicontinuous function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is the limit of a monotone decreasing sequence of extended real-valued continuous functions on ${\displaystyle X}$; if ${\displaystyle f}$ does not take the value ${\displaystyle \infty ,}$ the continuous functions can be taken to be real-valued.

• If ${\displaystyle C}$ is a compact space (for instance a closed bounded interval ${\displaystyle [a,b]}$) and ${\displaystyle f:C\to {\overline {\mathbb {R} }}}$ is upper semicontinuous, then ${\displaystyle f}$ has a maximum on ${\displaystyle C.}$ If ${\displaystyle f}$ is lower semicontinuous on ${\displaystyle C,}$ it has a minimum on ${\displaystyle C.}$

(Proof for the upper semicontinuous case: By condition (5) in the definition, ${\displaystyle f}$ is continuous when ${\displaystyle {\overline {\mathbb {R} }}}$ is given the left order topology. So its image ${\displaystyle f(C)}$ is compact in that topology. And the compact sets in that topology are exactly the sets with a maximum. For an alternative proof, see the article on the extreme value theorem.)

• Any upper semicontinuous function ${\displaystyle f:X\to \mathbb {N} }$ on an arbitrary topological space ${\displaystyle X}$ is locally constant on some dense open subset of ${\displaystyle X.}$
• Tonelli's theorem in functional analysis characterizes the weak lower semicontinuity of nonlinear functionals on L^p spaces in terms of the convexity of another function.

See also

• Directional continuity – Mathematical function with no sudden changesPages displaying short descriptions of redirect targets
• Katětov–Tong insertion theorem – On existence of a continuous function between semicontinuous upper and lower bounds
• Semicontinuous set-valued function

Notes

References

Bibliography

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