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Re: monotonic function
Posted:
Nov 29, 2012 5:47 AM
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On 11/29/2012 05:01 AM, David Bernier 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.
>
> 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\/
>
Upon further reflection, it seems that this paragraph here:
===================================================================
> 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.
===================================================================
can be omitted.
In the 1st case, there is a 'c' with a< c < b such that:
f(a), f(b) and f(c) are distinct and
f(c)> f(a) and f(c) > f(b).
In the 2nd case skectched above, there is once again a 'c'
with a< c < b , but in this instance, 'c' is such that:
f(a), f(b), f(c) distinct and
f(c) < f(a), and f(c) < f(b).
Them we can use the intermediate value theorem
on [a, c] as well as on [c, b], since f is
continuous:
>
> 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.
> --
>
> So, assuming f: R->R is continuous,
>
> f not strictly monotonic ==> f not injective.
>
> By the contrapositive, under the same assumptions on f,
> f injective ===> f strictly monotonic.
>
> On the other hand, assuming f: R->R is continuous,
> if f is strictly monotonic, a<b ==> f(a) < f(b) without
> loss of generality. That is seen to rule out f being
> non-injective.
>
> So on the other hand, assuming f: R->R is continuous,
> f strictly monotonic ==> f is injective.
>
> Taken together, we get:
> Under the assumption that f: R -> R is continuous, then:
> f is injective <==> f is strictly monotonic.
[...]
David Bernier
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