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Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from I_0 = [0,1]$I_0 = [0,1]$ by repeatedly removing the middle third of any ensuing interval. So let's denote by I_n$I_n$ the n$n$-th set in this process.

Now let's make a smooth function f_n$f_n$ on [0,1]$[0,1]$ such that its zero set is exactly I_n$I_n$. Starting with f_0 = 0$f_0 = 0$ we obtain f_{n+1}$f_{n+1}$ from f_n$f_n$ as follows:

set f_{n+1} = f_nSet $f_{n+1} = f_n$ on I_{n+1}$I_{n+1}$ and on an interval that is removed from I_n$I_n$ make f_{n+1}$f_{n+1}$ equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height 2^{-2^n}$2^{-2^n}$.

This choice of heights of the bump functions will ensure that the derivatives of f$f$ all converge uniformly to their pointwise limits. Hence the limit function f_n$f_n$ is again smooth. By construction its zero set is exactly the Cantor set.

Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from I_0 = [0,1] by repeatedly removing the middle third of any ensuing interval. So let's denote by I_n the n-th set in this process.

Now let's make a smooth function f_n on [0,1] such that its zero set is exactly I_n. Starting with f_0 = 0 we obtain f_{n+1} from f_n as follows:

set f_{n+1} = f_n on I_{n+1} and on an interval that is removed from I_n make f_{n+1} equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height 2^{-2^n}.

This choice of heights of the bump functions will ensure that the derivatives of f all converge uniformly to their pointwise limits. Hence the limit function f_n is again smooth. By construction its zero set is exactly the Cantor set.

Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from $I_0 = [0,1]$ by repeatedly removing the middle third of any ensuing interval. So let's denote by $I_n$ the $n$-th set in this process.

Now let's make a smooth function $f_n$ on $[0,1]$ such that its zero set is exactly $I_n$. Starting with $f_0 = 0$ we obtain $f_{n+1}$ from $f_n$ as follows:

Set $f_{n+1} = f_n$ on $I_{n+1}$ and on an interval that is removed from $I_n$ make $f_{n+1}$ equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height $2^{-2^n}$.

This choice of heights of the bump functions will ensure that the derivatives of $f$ all converge uniformly to their pointwise limits. Hence the limit function $f_n$ is again smooth. By construction its zero set is exactly the Cantor set.

corrected height of the bump
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Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from I_0 = [0,1] by repeatedly removing the middle third of any ensuing interval. So let's denote by I_n the n-th set in this process.

Now let's make a smooth function f_n on [0,1] such that its zero set is exactly I_n. Starting with f_0 = 0 we obtain f_{n+1} from f_n as follows:

set f_{n+1} = f_n on I_{n+1} and on an interval that is removed from I_n make f_{n+1} equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height 2^{-n2^n}.

This choice of heights of the bump functions will ensure that the derivatives of f all converge uniformly to their pointwise limits. Hence the limit function f_n is again smooth. By construction its zero set is exactly the Cantor set.

Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from I_0 = [0,1] by repeatedly removing the middle third of any ensuing interval. So let's denote by I_n the n-th set in this process.

Now let's make a smooth function f_n on [0,1] such that its zero set is exactly I_n. Starting with f_0 = 0 we obtain f_{n+1} from f_n as follows:

set f_{n+1} = f_n on I_{n+1} and on an interval that is removed from I_n make f_{n+1} equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height 2^{-n}.

This choice of heights of the bump functions will ensure that the derivatives of f all converge uniformly to their pointwise limits. Hence the limit function f_n is again smooth. By construction its zero set is exactly the Cantor set.

Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from I_0 = [0,1] by repeatedly removing the middle third of any ensuing interval. So let's denote by I_n the n-th set in this process.

Now let's make a smooth function f_n on [0,1] such that its zero set is exactly I_n. Starting with f_0 = 0 we obtain f_{n+1} from f_n as follows:

set f_{n+1} = f_n on I_{n+1} and on an interval that is removed from I_n make f_{n+1} equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height 2^{-2^n}.

This choice of heights of the bump functions will ensure that the derivatives of f all converge uniformly to their pointwise limits. Hence the limit function f_n is again smooth. By construction its zero set is exactly the Cantor set.

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Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from I_0 = [0,1] by repeatedly removing the middle third of any ensuing interval. So let's denote by I_n the n-th set in this process.

Now let's make a smooth function f_n on [0,1] such that its zero set is exactly I_n. Starting with f_0 = 0 we obtain f_{n+1} from f_n as follows:

set f_{n+1} = f_n on I_{n+1} and on an interval that is removed from I_n make f_{n+1} equal to a bump function that is 0 only at the boundary of the interval. We can choose the bump function to be of height 2^{-n}.

This choice of heights of the bump functions will ensure that the derivatives of f all converge uniformly to their pointwise limits. Hence the limit function f_n is again smooth. By construction its zero set is exactly the Cantor set.