Is there is a non-constant function such that each point is its local minimum?

Asked Jul 09 '19 at 02:24

Active Jul 09 '19 at 03:12

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Is there is a non-constant real-valued function on $\mathbb{R}$ such that each point is its local minimum, i.e. its value at any point is no greater than any other value in a sufficiently small neighborhood of the point. The answer is no if continuity is assumed, see Continuous functions that attain local extrema at every point.

If there is such a function, is its range necessarily a countable set or not?

edited Jul 09 '19 at 03:12

Eric Wofsey

330,363

asked Jul 09 '19 at 02:24

Mingwei Zhang

A constant. ${}{}$ – copper.hat Jul 09 '19 at 02:29
If each point is a minimum then $f(x_0)\leq f(x)$ for arbitrarily chosen $x_0$ and $x$. Switching the roles of $x$ and $x_0$, also $f(x)\leq f(x_0)$. Hence $f(x)=f(x_0)=\textrm{const}$. – Conifold Jul 09 '19 at 02:29
What is your definition of a minimum point? Do all the nearby function values need to be strictly greater or just no less than the value at the point in question. In the second case, a constant function suffices. Your idea should have suggested the constant function as something of interest. Why didn't you pursue the idea? – Ross Millikan Jul 09 '19 at 02:29
@RossMillikan just the 2nd case – Mingwei Zhang Jul 09 '19 at 02:32
@Conifold You can’t simply switch them, as each point is LOCAL minimum point. – Mingwei Zhang Jul 09 '19 at 02:34
@MingweiZhang The OP does not say anything about local. – Conifold Jul 09 '19 at 02:35
@Conifold Sorry for that. I’ll add it. – Mingwei Zhang Jul 09 '19 at 02:36
You'll get that it is locally constant by the same argument. As long as the space (I guess $\mathbb{R}$) is connected, locally constant implies globally constant by a continuity argument. There'd be a sequence of points with one value converging to a point with a different value otherwise, and it won't be locally constant at the limit point. – Conifold Jul 09 '19 at 02:40
@Conifold I believe this argument is right, but still don’t know how to get local constant. – Mingwei Zhang Jul 09 '19 at 02:45
I am not so sure anymore. Typical arguments use continuity at some point, see Continuous functions that attain local extrema at every point. There might be some pathological counterexamples. – Conifold Jul 09 '19 at 02:49

Is there is a non-constant function such that each point is its local minimum?

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