Let X be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and F:X*[0,1]->R a continuous function such that F(x,0)>0 for all x in X F(x,1)<0 for all x in X Then, for each x in X, let t_0(x) in [0,1] be the smallest such that F(x,t_0(x))=0. Is t_0:X->[0,1] a continuous function?

Jenny Stafford

Jenny Stafford

Open question

2022-08-19

Let X be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and F : X × [ 0 , 1 ] R a continuous function such that

1. F ( x , 0 ) > 0 for all x X
2. F ( x , 1 ) < 0 for all x X

Then, for each x X, let t 0 ( x ) [ 0 , 1 ] be the smallest such that F ( x , t 0 ( x ) ) = 0. Is t 0 : X [ 0 , 1 ] a continuous function?

A friend suggested that I applied the Maximum theorem, but to show that the relevant correspondence is lower semicontinuous I need to prove the following statement:

If x n x and F ( x , t ) = 0, there is a subsequence x n k and a sequence t k such that F ( x n k , t k ) = 0 and t k t. This doesn't seem very obvious or even true, but I'm not sure.

Any suggestions?

Answer & Explanation

Kelsie Marks

Kelsie Marks

Beginner2022-08-20Added 17 answers

Let X also be some interval, e.g., [ 1 , 1 ], and let
F ( x , t ) = x 2 ( 1 2 t )
Then t ( x ) = 1 / 2 for all x 0 while t ( 0 ) = 0, thus no continuity at zero.

One can add a slight perturbation so that the set F ( 0 , t ) = 0 is some smaller segment, e.g., [ 0.2 , 0.8 ] and the conditions are also satisfied at x = 0.

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