Instead, uniform continuity can be defined on a metric space where such comparisons are possible, or more generally on a uniform space.
12.
Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed interval is uniformly continuous.
13.
*PM : uniform continuity of Lipschitz functions, id = 6834-- WP guess : uniform continuity of Lipschitz functions-- Status:
14.
*PM : uniform continuity of Lipschitz functions, id = 6834-- WP guess : uniform continuity of Lipschitz functions-- Status:
15.
Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.
16.
The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasn't published until the 1930s.
17.
Moduli of continuity are mainly used to give a quantitative account both of the continuity at a point, and of the uniform continuity, for functions between metric spaces, according to the following definitions.
18.
In the case " p " = " the above property does not hold in general : actually, it exactly reduces to the uniform continuity, and defines the uniform continuous functions.
19.
When " X " is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces.
20.
This naturally resulted in a number of contradictions and discrepancies in individual characters'histories, so a new, uniform continuity was created and the revised origins of the resulting heroes were retold in the hopes of maintaining consistent continuity.
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