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Re: monotonic function
Posted:
Nov 29, 2012 12:15 PM
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On Thu, 29 Nov 2012 05:01:18 -0500, David Bernier
<david250@videotron.ca> 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.
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