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2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all x. I(x)h(x)c( )exp 0 @ X i=1 w i( )t i(x) 1 Adx= @logc( ) @ j Z x h(x)c( )exp 0 @ Xk i=1 w i( )t i(x) 1 Adx Or E 0 B @ Xk i=1 @w i( ) @ j t i(x) 1 C A= @ @ j logc( ) To prove statement (b) of the theorem di erentiate a second time and rearrange Note can be a single parameter or a vector of parameters Show that X˘( ;. A w a ő i e ł l ̋k ͂ ւ i k x h vietnamryukyu culturaltechnological village @ x g i h Њt v b h b v.

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