I have f: (0,1] -> R defined by f(x)= xsin(1/x) where 0 < x =< 1.

I have to show that it is uniformly continuous on it's domain.

I know the definition of uniform continuity is

f is uniformly continuous on (0,1] if for all E>0 there exists a D>0 such that d(x,y)<D => d(f(x), f(y)) <E

I know what the graph of this function looks like, and I can see that it is continuous on (0,1]

I'm not sure what x and y to pick to prove the continuity. I think I should use something to do with (0,1] being a(This is not compact, it is not closed)? Thankscompact set

So take