Hemicontinuity

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In mathematics, the concept of continuity as it is defined for single-valued functions is not immediately extendible to multi-valued mappings or correspondences. In order to derive a more generalized definition, the dual concepts of upper hemicontinuity and lower hemicontinuity are introduced.

1 Upper hemicontinuity
2 Lower hemicontinuity
3 Implications for continuity
4 See also
5 References

[edit] Upper hemicontinuity

A correspondence Γ : A → B is said to be upper hemicontinuous if the images of all compact sets are bounded, and:

$\forall a \in A, a^m \rarr a, b^m \in \Gamma(a^m),$ and $b^m \rarr b \, \implies b \, \in \Gamma(a)$

In words, this definition simply says that a correspondence is upper hemicontinuous if and only if it has a closed graph and the image of compact sets is bounded.

[edit] Lower hemicontinuity

A correspondence Γ : A → B is said to be lower hemicontinuous at the point a if the images of all compact sets are bounded, and:

for each convergent sequence in the domain $a^m \rarr a$ , and each value in the range ( $b \in \Gamma(a)$ ), there exists a sequence such that ( $b^m \rarr b$ ). The sequence is such that the mth point of the sequence in the range ( $b m$ ) is in the range of the mth point in the sequence in the domain ( $a m$ ), for all $m > M$ for some value of M.

Or, stated entirely in set notation,

$\forall a^m \rarr a (a, a^m \in A), \forall b \in \Gamma(a) : \exists M, \exists b^m \rarr b$ such that $b^m \in \Gamma(a^m) (m>M)$

[edit] Implications for continuity

If a correspondence is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous. A continuous function is in all cases both upper and lower hemicontinuous.