Transcription of Continuity and Uniform Continuity
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Continuity and Uniform Continuity
Continuity and Uniform Continuity521 May 12, denote a subset of the real numbersRandf:S Rwill be a real valued function defined onS. The setSmay be bounded likeS= (0,5) ={x R: 0< x <5}or infinite likeS= (0, ) ={x R: 0< x}.It may even be all ofR. The valuef(x) of the functionfat the pointx Swill be defined by a formula (or formulas).Definition functionfis said to becontinuous onSiff x0 S >0 >0 x S[|x x0|< = |f(x) f(x0)|< ].Hencefis not continuous1onSiff x0 S >0 >0 x S[|x x0|< and|f(x) f(x0)| ].Definition functionfis said to beuniformly continuous onSiff >0 >0 x0 S x S[|x x0|< = |f(x) f(x0)|< ].Hencefis not uniformly continuous onSiff >0 >0 x0 S x S[|x x0|< and|f(x) f(x0)| ].1 For an example of a function which isnotcontinuous see Example 22 only difference between the two definitions is the order of the quan-tifiers.
for x 1;x 2 aas follows. First a2 x 1x 2 since a x 1 and a x 2.Then jx 1 1 x 1 2 j= jx 1 x 2j x 1x 2 jx 1 x 2j a2 (2) where we have used the fact that 11 < if 0 < < . It follows that that the function f(x) is uniformly continuous on any interval (a;1) where
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