Wikipedia:Reference desk/Archives/Mathematics/2023 June 23

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June 23

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Is there any general non-obvious sufficient condition, for a given differentiable injection   to satisfy that   (for all  ) is not injective?

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Here are some simple examples of a differentiable injection  , for which   (for all  ) is not injective:

  1.   defined on the set of real numbres, for any real  
  2.   defined on the set of real numbres, for any real   and for any odd number   and for any real  
  3.   defined on the set of positive numbres, for any real   and for any natural   and for any positive  
  4.   defined on the set of positive numbres, for any positive base  
  5.   defined on the interval   for any real  

By a "general" sufficient condition, I exclude any of the sufficient examples mentioned above. They are really sufficient (and non-obvious), yet not general, but rather special cases.

By a "non-obvious" sufficient condition, I exclude any obvious sufficient condition like the following (general) one: "  for some differentiable function   that is not injective while   is".

2A06:C701:427F:6800:8CE8:BDC9:AFA0:45F4 (talk) 09:26, 23 June 2023 (UTC)[reply]

Few notes to make:
1. Assuming that   is  ,   exists either properly or as a limit across the real line if and only if  . The only if direction is obvious since   would make   not exist. The if direction results from L'Hôpital's rule, guaranteeing that  .
2. If we do assume that  , and if we furthermore assume that   is  , the derivative of   exists everywhere (again either properly or as a limit), as by L'Hôpital's rule,  .
3. In general, assuming that   is differentiable,   is also injective if and only if   is always nonnegative or nonpositive, and only   at isolated points (I will call this property A.)
4. Combined together, if we assume that   is a   injection with  , then it really boils down to   having property A, but   not having property A.
GalacticShoe (talk) 16:59, 23 June 2023 (UTC)[reply]
Few notes to make:
  • Thanks ever so much.
  • Re. the end of your first section: By the last mathematical expression   appearing to the right of the identity sign, you have probably meant   haven't you?
  • Re. the end of your second section: By the last mathematical expression   appearing to the right of the identity sign, you have probably meant   haven't you?
  • Re. your fourth section: Even though your general sufficient condition does not cover my fourth example above (because the logarithmic function is not defind for zero), nor does it cover my third example (for a similar reason), your condition is still a general (non-obvious) sufficient one, as required (BTW, practically speaking, I need all of this for functions not defined for zero. I forgot to mention that in my original question).
2A06:C701:7453:7D00:8F6:E1A9:A503:D522 (talk) 19:44, 24 June 2023 (UTC)[reply]
In regards to the second and third question, that's my mistake; it should be   and  , I will edit my answer accordingly. GalacticShoe (talk) 20:59, 24 June 2023 (UTC)[reply]
You should have also added the condition that   is continuous, shouldn't you? 2A06:C701:7453:7D00:8F6:E1A9:A503:D522 (talk) 21:09, 24 June 2023 (UTC)[reply]
Good point, let me change to continuous differentiability to make it work in general. GalacticShoe (talk) 22:29, 24 June 2023 (UTC)[reply]
You may want to precede 'lim' with a backslash. This would make the 'lim' a symbol, rendered in the upright font with appropriate spacing, instead of a blob of italic, varable-like letters. Compare \lim x  to lim x . --CiaPan (talk) 20:24, 25 June 2023 (UTC)[reply]
Will do, thanks for the heads up! GalacticShoe (talk) 00:19, 26 June 2023 (UTC)[reply]