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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)|< ].

The function fis said to be uniformly continuous on Si 8">0 9 >0 8x 0 2S8x2S jx x 0j< =)jf(x) f(x 0)j<" : Hence fis not uniformly continuous on Si 9">0 8 >0 9x 0 2S9x2S jx x 0j< and jf(x) f(x 0)j " : 1For an example of a function which is not continuous see Example 22 below. 1. 4. The only di erence between the two de nitions is the order of ...

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people.math.wisc.edu

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