## Fun with restrictions

If $f: A \mapsto B$ is a function, let us call $f: a\subset A \mapsto B$ a restriction of $f$ to $a$. Let’s analyze one nice and pretty surprising fact concerning restrictions.

Fact: Suppose we have a function of two variables $f(x, y)$ and it’s restriction to every straight line going through the origin $f(l_1t, l_2t)$ has a strict local minimum at $t=0$. This does not necessarily imply that $f$ has a local minimum at the origin.

Funny, isn’t it – all the (straight) roads lead down, and yet you are not on top.

Counterexample: $z=(x^2-y)(y-x^2)$. Verifying that a) this function does not have a local minimum at $[0, 0]$ and b) $(l_1^2t^2-l_2t)(l_2t-2l_1^2t)$ has a local minimum at $t=0$ is trivial, boring, and will be left as an exercise for the reader.

What’s going on here? Let’s plot this function:

Well, maybe this time it doesn’t help much. But if we look at this surface above, it will make things more clear:

What we have here is a kind of a distorted saddle at the origin (big black point). The function is equal to zero there. There are points where the function is positive (red region) in every neighborhood of the origin, no matter how small, so it’s definitely not a local maximum. But at the same time, every straight line going through the origin will have to cross the blue region (where the function if negative) before it crosses  the red region, and therefore the restriction of a function to that line will have to decrease for a little while and will have a local minimum. No matter which straight line you chose, you can’t get to the red region before you get to the blue one. Now, if you were allowed to choose a parabola…

But even then I could modify the example so that even parabolic roads wouldn’t help. I think you can now guess how – by replacing squares with 4-th powers: $z=(x^4-y)(y-x^4)$. Now, we could take not one, but two families of lines, each completely covering the plane – all the straight lines $x=l_1 t, y=l_2 t$ and all the parabolas $x=t, y=ct^2$, and still all the restrictions would have strict local maxima, while the function itself wouldn’t. (We could take this even further, by using such functions as $exp(1/x^2)$, but then  we would have to dance the singularity-patching dance, and that’s boring.)