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

= June 23 =

Is there any general non-obvious sufficient condition, for a given differentiable injection $$f(x)$$ to satisfy that $$f(x)/x$$ (for all $$x \ne 0$$) is not injective?
Here are some simple examples of a differentiable injection $$f$$, for which $$f(x)/x$$ (for all $$x \ne 0$$) is not injective:

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.
 * 1) $$f: x\mapsto A\cdot x,$$ defined on the set of real numbres, for any real $$A \ne 0.$$
 * 2)  $$f: x\mapsto A\cdot x^B+C,$$ defined on the set of real numbres, for any real $$A \ne 0,$$ and for any odd number $$B \ne 1,$$ and for any real $$C.$$
 * 3)  $$f: x\mapsto A\cdot x^B+C,$$ defined on the set of positive numbres, for any real $$A \ne 0,$$ and for any natural $$B \ne 1,$$ and for any positive $$C$$
 * 4) $$f: x\mapsto \log_{B}(x),$$ defined on the set of positive numbres, for any positive base $$B \ne 1.$$
 * 5) $$f: x\mapsto A\cdot\arcsin\sqrt{x},$$ defined on the interval $$[0,1],$$ for any real $$A \ne 0.$$

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

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


 * Few notes to make:
 * 1. Assuming that $$f(x)$$ is $$C^{1}$$, $$f(x)/x$$ exists either properly or as a limit across the real line if and only if $$f(0) = 0$$. The only if direction is obvious since $$f(0) \neq 0$$ would make $$\lim_{x \rightarrow 0} f(x)/x$$ not exist. The if direction results from L'Hôpital's rule, guaranteeing that $$\lim_{x \rightarrow 0} f(x)/x = f'(0)$$.
 * 2. If we do assume that $$f(0) = 0$$, and if we furthermore assume that $$f$$ is $$C^{2}$$, the derivative of $$f(x)/x$$ exists everywhere (again either properly or as a limit), as by L'Hôpital's rule, $$\lim_{x \rightarrow 0} (xf'(x)-f(x))/x^{2} = f''(0)/2$$.
 * 3. In general, assuming that $$g(x)$$ is differentiable, $$g(x)$$ is also injective if and only if $$g'(x)$$ is always nonnegative or nonpositive, and only $$0$$ at isolated points (I will call this property A.)
 * 4. Combined together, if we assume that $$f(x)$$ is a $$C^{2}$$ injection with $$f(0) = 0$$, then it really boils down to $$f'(x)$$ having property A, but $$xf'(x) - f(x)$$ not having property A.
 * GalacticShoe (talk) 16:59, 23 June 2023 (UTC)
 * Few notes to make:
 * Thanks ever so much.
 * Re. the end of your first section: By the last mathematical expression $$f'(x),$$ appearing to the right of the identity sign, you have probably meant $$\lim_{x \rightarrow 0} f'(x),$$ haven't you?
 * Re. the end of your second section: By the last mathematical expression $$f(x)/2,$$ appearing to the right of the identity sign, you have probably meant $$\lim_{x \rightarrow 0} f(x)/2,$$ 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)
 * In regards to the second and third question, that's my mistake; it should be $$f'(0)$$ and $$f''(0)/2$$, I will edit my answer accordingly. GalacticShoe (talk) 20:59, 24 June 2023 (UTC)
 * You should have also added the condition that $$f'$$ is continuous, shouldn't you? 2A06:C701:7453:7D00:8F6:E1A9:A503:D522 (talk) 21:09, 24 June 2023 (UTC)
 * Good point, let me change to continuous differentiability to make it work in general. GalacticShoe (talk) 22:29, 24 June 2023 (UTC)
 * 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 → $$ \lim x$$ to lim x → $$lim x$$. --CiaPan (talk) 20:24, 25 June 2023 (UTC)
 * Will do, thanks for the heads up! GalacticShoe (talk) 00:19, 26 June 2023 (UTC)