Abstract ▼ AI-Helper

Let f be defined on a subset D(f)of R^(p) to R^(q).Then it is readily seen that the following statements are equivalent: ⅰ) f is continuous at every point in D(f) ⅱ) Given >0 and u∈ D(f),there in a δ( ,u)>0 such that if x belongs to D(f)and ||x - u||≤ δ,then ||f(x)- f(u)||≤ . The fact to be noted he...

Let f be defined on a subset D(f)of R^(p) to R^(q).Then it is readily seen that the following statements are equivalent: ⅰ) f is continuous at every point in D(f) ⅱ) Given >0 and u∈ D(f),there in a δ( ,u)>0 such that if x belongs to D(f)and ||x - u||≤ δ,then ||f(x)- f(u)||≤ . The fact to be noted here is that δ depends,in general,on both and u.Deperdence of δ depends on u is a reflection of the fact that the function f may change its values rapidly near certain points and slowlynearothers. Now it can happen that a function is such that the number δ can be chosen to be independent of the point u in D(f) and depending only on . In this thesis we study continuous functions and their applications. The first chapter includes introductory remarks.In the second chapter. I introduce the definitions and basic properties of the limits of the limits of functions and functions of several kindsofcontinuity such as semi-continuity,uniform continuity,and absolutely continuity in R. In the third chapter,the domains are generalized form the ones in R to the ones in R^(n).Finally,we study the Stone-Weierstrass Theorem using the Stone Approximation Theorem,the TietzeExtension Theorem using the continuity ofthe given function on a closed subset of R^(n) and the Arzela - Ascoli Theorem using the Bolzano-Weierstrass Theorem.

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