Give an example of a function f: (0,1) -> R that is differentiable and uniformly continuous on (0,1) but such that f' is unbounded.
2006-11-10 07:30:12 · 3 answers · asked by Doug A 2 in Science & Mathematics ➔ Mathematics
So are you looking for a function with a singularity on (0,1)? That would address the question I think.
maybe (x-(1/2))^(1/2)
THE FIRST ANSWER IS WRONG, I THINK, BECAUSE YOUR DOMAIN IS OPEN AND DOESN'T INCLUDE 0.
2006-11-10 07:43:54 · answer #1 · answered by Anonymous · 0⤊ 0⤋
f(x)=1/x is not uniformly continuous on (0,1). The limit of a uniformly continuous function at a limit point will always exist. However,
f(x)=sqrt(x) is uniformly continuous and f'(x) is unbounded there.
2006-11-10 07:45:02 · answer #2 · answered by mathematician 7 · 1⤊ 0⤋
f(x) = 1/x. It's differentiable and uniformly continuous on (0,1), but at x = 0, f' is infinite.
2006-11-10 07:42:54 · answer #3 · answered by Scythian1950 7 · 0⤊ 2⤋