On Thu, 29 Nov 2012 05:01:18 -0500, David Bernier <email@example.com> wrote:
>On 11/28/2012 09:27 AM, Timothy Murphy wrote: >> quasi wrote: >> >>> Any increasing function is monotonic. >>> >>> Any decreasing function is monotonic. >> ... >> >> Also, maybe: a continuous function f: R->R is strictly monotonic >> if and only if it is injective? >> > >My deductions show that: > > >If a continuous function f: R -> R isn't strictly monotonic, >it might or might not be injective.
What's an example of a continuous f : R -> R which is not strictly monotonic but is injective?
> >If f is not injective, we're done. >-- > > > > >Then the remaining case is where f is injective: > >f not strictly monotonic and yet f injective over an >interval [a, b] with a< b would give one of two graph shapes: > >(f continuous) > >x = a/\ > \x = b > >or: > /x = b >x=a\/ > >There's a strict maximum at some point between a and b (1st case), >and a strict minimum of f at some point between a and b (2nd case). >This is because f is continuous and [a, b] is a compact interval. > >Then, by the intermediate value theorem (whether it be the 1st or >2nd case), it's easy to see that some value in the range of >f on [a, b] is attained at at least two distinct points >in [a, b]. So f is not injective on [a, b]. >So f: R -> R isn't injective. >This contradicts the assumption that f is injective.