Introduction To Analysis Ii Show All Steps Definition 5 4 1 1 (19.5 KiB) Viewed 29 times
Definition 5.4.1
Introduction To Analysis Ii Show All Steps Definition 5 4 1 2 (28.77 KiB) Viewed 29 times
1. Prove that each function is uniformly continuous on the given set by directly verifying the E - 8 property in Definition 5.4.1. (a) f(x) = us on (0,2] 1 (b) f(x) on (2,00) 2 (c) f(x) = 2-1 2+1 on (0,0)
4.1 DEFINITION Let f:D R. We say that f is uniformly continuous on Dif for every e > 0 there exists a 8 >0 such that Sx)-f()<E whenever |x - yl < 8 and x, ye D. It should be clear that if a function is uniformly continuous on a set D, then it is certainly continuous on D. Furthermore, while it is proper to speak of a function being continuous at a point, uniform continuity is a property that applies to a function on a set. We never speak of a function being uniformly continuous at a point.
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