Formal Definition
Let f be a complex-valued function defined on a measure space, (X, μ) that also belongs to L∞(μ). Then the essential range of f is defined to be the set:
- S = { complex numbers z | μ({ x : abs(f(x) − z) < ε }) > 0 for all ε > 0}
Note that: Another description of the essential range of a function is as follows:
The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.
The above description of the essential range is equivalent to the formal definition of the essential range and will therefore be used throughout this article.
Read more about this topic: Essential Range
Famous quotes containing the words formal and/or definition:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)