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.
as required. (Note that x 0 is large when is small.) 11. According to the Mean Value Theorem from calculus for a di erentiable function fwe have f(x 1) f(x 2) = f0(c)(x 2 x 1): for some cbetween x 1 and x 2. (The slope (f(x 1) f(x 2))=(x 1 x 2) of the secant line joining the two points (x 1;f(x 1)) and (x 2;f(x 2)) on the graph is
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