Hello... i was trying to do this but stuck in between ...can anyone help me?
F (x)= xsin (1/x): x not equal to 0
F (x)= 0 : for x=0
For all x belongs closed interval -1,1..
Prove f (x) is uniformly continuous by using € defination?
A Euro definition? Do you have to buy it? I assume you mean the $\displaystyle \delta-\epsilon$ definition: a function, f, is "uniformly continuous" on set A if and only if, for every $\displaystyle \epsilon> 0$ there exist $\displaystyle \delta> 0$ such that if for all $\displaystyle x\in A$ if $\displaystyle |x- a|<\delta$ then $\displaystyle |f(x)- f(a)|< \epsilon$.
(The difference between "continuous" and "uniformly continuous" is the if a function is uniformly continuous on set A, we can choose a $\displaystyle \delta$ that works for all a in A. If the function is only continuous, possible values for $\displaystyle \delta$ may depend both $\displaystyle \epsilon$ and a.)