Prove that each of the following functions is uniformly continuous on (0,1). (You may use l'Hopital's Rule).
b)f(x) = xcos(1/x^2)
attemp in proof: I need to use the following theorem. Theorem 3.40: Suppose that a < b and that f:(a,b) → R. Then f is uniform continuous on (a,b) if and only if f can be continuously extended to [a,b]; that is, if and only if there is a continuous function g:[a,b] → R which satisfies f(x) = g(x), where x is an element of (a,b).
Then cos(1/x^2) is bounded for all x in (0,1) not including 0. And x → 0 as x → 0, then f(x) → 0 as x → 0. Thus, the limit exists as an extended real number, so using the above theorem, if 0 < 1, and suppose f:(0,1) → R, then f is uniform continuous on (0,1) if and only if xcos(1/x^2) can be continously extended to [0,1].
Can someone please help me? I don't know how or why to conclude with theorem 3.40. Thank you for any help.
You are finished. Let $g(x)=f(x)$ when $x\ne 0$, and let $g(0)=0$. Then $g$ is continuous on $[0,1]$.