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

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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 $x\neq 0$) is not injective?

Here are some simple examples of a differentiable injection $f$, for which $f(x)/x$ (for all $x\neq 0$) is not injective:

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

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: "$f:x\mapsto xg(x)$ 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 → $limx$. --CiaPan (talk) 20:24, 25 June 2023 (UTC)

Will do, thanks for the heads up! GalacticShoe (talk) 00:19, 26 June 2023 (UTC)