Continuity and Uniform Continuity - University of …

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 same as the slope f0(c) of the tangent point at the intermediate point (c;f(c)).) If x 1 and x 2 lie in some interval Sand jf0(c)j Mfor all c2S we conclude that the Lipschitz inequality (1) holds on S. We don’t want to use the Mean Value Theorem without rst proving it, but we certainly can

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  Value, Intermediate, Theorem, Value theorem, The intermediate

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