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)|< ].

13.p We can discover a Lipscitz inequality for the square root function f(x) = xin much the same way. Consider the function f(x) = p xde ned on the interval S= (a;1) where a>0. For x 1;x 2 2Sthe Mean Value Theorem says that p x 1 p x 2 = (x 1 x 2)=(2 p c) where cis between x 1 and x 2. If x 1;x 2 2S then c 2S (as cis between x 1 and x 2) and ...

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