lower semicontinuous function

lower semicontinuous function

[¦lō·ər ‚sem·ē·kən′tin·yə·wəs ‚fənk·shən]
(mathematics)
A real-valued function ƒ(x) is lower semicontinuous at a point x0 if, for any small positive number ε, ƒ(x) is always greater that ƒ(x0) - ε for all x in some neighborhood of x0.
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where f : E [right arrow] E is a self mapping and [phi] : [0, + [infinity]) [right arrow] [0, + [infinity]) is a lower semicontinuous function from right such that [phi] is positive on (0, + [infinity]) and [phi](0) = 0.
and also [phi]: [0,+[infinity]) [right arrow] [0,+[infinity]) is lower semicontinuous function from right such that [phi](t) > 0 for t > 0 and [phi](0) = 0.
and moreover [phi] : [0,+[infinity]) [right arrow] [0,+[infinity]) is lower semicontinuous function from right such that [phi](t) > 0 for t > 0 and [phi](0) = 0.