- #1
math8
- 160
- 0
(R:reals)
Let f:R-->R be continuous and satisfy |f(x)-f(y)|>or eq. to k|x-y| for all x, y in R and some k>0. Show that f is surjective.
I can show that f is injective: let f(x) = f(y), hence k|x-y|< or eq. to 0, thus x=y.
I had a suggestion that it might be helpful to show that f has closed image. But I don't see how to work with that.
I know that f is surjective if for each y in R there is an x in R such that f(x)=y.
I don't see how to proceed.
Let f:R-->R be continuous and satisfy |f(x)-f(y)|>or eq. to k|x-y| for all x, y in R and some k>0. Show that f is surjective.
I can show that f is injective: let f(x) = f(y), hence k|x-y|< or eq. to 0, thus x=y.
I had a suggestion that it might be helpful to show that f has closed image. But I don't see how to work with that.
I know that f is surjective if for each y in R there is an x in R such that f(x)=y.
I don't see how to proceed.