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Let E be a closed set in the real number R and a real valued and continuous function f be given on R. What are the examples satisfying the following conditions?

  1. The function f is differentiable on R/ n times differentiable R/or has derivatives of all orders on R.
  2. The zero set of f is E. (i.e. E={p | f(p)=0})

If not exist, then prove it why the case does not exist.

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Hints: Examples do exist for both of these.

1) Try some very simple things.

2) The easiest thing to do would be to have the function such that $f(x)=0$ when $x\in E$ and $f(x)=1$ when $x\notin E$, unfortunately this is not continuous. Notice the discontinuities come from the ''boundary'' of $E$. If you could "lift" the function over finite distance instead of zero distance, you could remove the discontinuity. How could you do this, intuitively? You might imagine this could fail for some very pathological set $E$; why is this always possible?

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  • $\begingroup$ "If you could lift the fuction ~... discontinuity" : I don't understand this sentence... $\endgroup$ – user94217 Sep 11 '13 at 13:35
  • $\begingroup$ Perhaps a more physical analogy is in order. Sometimes in very early physics classes we pretend that a car can have its velocity change from zero to (say) 10 m/s instantaneously. Of course, this is not true, in reality it will take some time no matter how good your engine is. Our naïve $f$ is suffering from the same problem as that discontinuous velocity function, and the fix is the same as well. $\endgroup$ – Eric Stucky Sep 11 '13 at 20:58

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