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 that1. F ( x , 0 ) &gt; 0 for all x ∈ X2. F ( x , 1 ) &lt; 0 for all x ∈ XThen, 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?

Question

Let   X be a compact metric space (feel free to impose more conditions as long as they&#039;re also satisfied by spheres) and   F  :  X  ×  [  0  ,  1  ]  →      R   a continuous function such that1.   F  (  x  ,  0  )  &amp;gt;  0 for all   x  ∈  X2.   F  (  x  ,  1  )  &amp;lt;  0 for all   x  ∈  XThen, 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&#039;t seem very obvious or even true, but I&#039;m not sure.Any suggestions?

Kelsie Marks · Accepted Answer

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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