Real analysis: inequality on minimum points

Question

Let $f,g$ be two real functions. I wonder if the following holds true: $$\lvert\arg \min_x f(x)-\arg \min_x g(x)\rvert\leq \arg\min_x\lvert f(x)-g(x) \rvert.$$

Do you have any conclusion about it? Are there related results on this?

Thanks in advance

Answer 1

If I just say the answer is "no", then that's really misleading as you might think that inequality makes sense in general when it doesn't. Actually $\mathrm{arg} \min\,f$ is a set, and the inequality, as it is, doesn't make much sense. Only if we assume ALL those functions reach their min at one single point then we can make some sense of it, by identifying arg$\min\, f$ with number it contains. It's still wrong though, as the following example shows.

Take $f(x)=\cos x$ and $g(x)=\sin x$ in the interval $[0,\pi]$. Then $\mathrm{arg}\, \min f= \pi$ and $\mathrm{arg}\, \min g= 0$ while $\mathrm{arg}\, \min |\cos x - \sin x |= 3\pi/4$