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equicontinuous
equicontinuous adj. 
/ˌɛkwᵻkənˈtɪnjʊəs/, /ˌɛkwikənˈtɪnjuəs/, /ˌikwəkənˈtɪnjuəs/, /ˌikwikənˈtɪnjuəs/ Mathematics of a set of functions fi(x): having the property that for all x1 and all ε > o there corresponds a δ > o such that, if |x − x1| < δ, |fi(x) − fi(x1) | < ε for all i. [Translating Italian egualmente continuo (C. Arzelà 1895, in Mem. d. R. Accad. d. Sci. dell'Ist. di Bologna V. 226).]
Brit. /ˌiːkwᵻkənˈtɪnjʊəs/
, /ˌiːkwᵻkənˈtɪnjʊəs//ˌɛkwᵻkənˈtɪnjʊəs/U.S. /ˌɛkwəkənˈtɪnjuəs/
, /ˌɛkwəkənˈtɪnjuəs//ˌɛkwikənˈtɪnjuəs//ˌikwəkənˈtɪnjuəs//ˌikwikənˈtɪnjuəs/ΚΠ
1926 E. W. Hobson Theory of Functions of Real Variable (ed. 2) II. ii. 168 In case the family of continuous functions is such that, for each value of ε, a single set of sub-intervals or sub-cells can be so determined that, for every function f(x) of the family, the fluctuation in a sub-interval, or a sub-cell, of a set, is less than ε, the family is said to consist of equi-continuous functions.
1959 G. James & R. C. James Math. Dict. (ed. 2) 21/1 Ascoli's theorem. From any set of uniformly bounded functions equicontinuous on a bounded closed (compact) set (such as a closed interval) it is possible to select an infinite sequence which converges uniformly to a limit function which is also continuous.
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