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### Finding a $\text C^\infty$ function which is zero just on a given closed set [duplicate]

If $\text E$ is an arbitrary closed set in $\mathbf R^n$, show that there is an $\text f\in \text C^\infty(\mathbf R^n)$ such that $\text f(x)=0~$for every $x \in \text E$ and $\text f(x)>0$ for ...
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### There is any smooth function on real line which is zero just in [0,1] [duplicate]

There is any function of class $C^{\infty}$ with the following property: $f:\mathbb{R}\to \mathbb{R}$ $$f(x)=0\ \ \ \ \ \text{iff} \ \ \ \ \ \ 0\leq x\leq 1?$$ I am wondering because of the ...
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### What does a topology do, and what makes a particular topology the 'right' one?

From Wikipedia: The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies. [Aside: ...
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### Other any other versions of identity theorem.

The classical identity theorem states that: If $f(x)$ a real analytic function on a domain $D \subset \mathbb{R}$. Suppose $f(x)=0$ on some $M \subset D$ such that $M$ has an accumulation ...
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### Zero-sets of BV functions

Let $f\colon [0,1]\to \mathbb{R}$ be a function of bounded variation. As $f$ is the difference of two monotone functions, the zero set $X = f^{-1}(0)$ is Borel (can we say more?). Can we find a BV ...
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### Is this statement about manifold true? [duplicate]

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
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### Boundary of the set of critical points

Let $u\in C^\infty(\mathbb{R}^d)$ and consider $$E = \partial\{x:Du(x)=0\}.$$ I am interested in understanding whether $\mu(E)=0$ or not, $\mu$ being the $d$-dimensional Lebesgue measure. What I ...
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### $V$ is open , then $V=\{x\in \mathbb R:f(x)>0\}$ for some continuous function $f$

Let $V$ be a non-empty open set of real numbers , then how to prove that there is a continuous function $f:\mathbb R\to \mathbb R$ such that $V=\{x\in \mathbb R:f(x)>0\}$
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### Inner point of zero set with positive measure

Let $f:B_r(0)\rightarrow\mathbb{R}$ with $B_r(0)\subset \mathbb{R}^2$ the open Ball of radius $r>0$. Let us further denote the set of all zeros of $f$ by $$M:=\{x\in B_r(0):\ f(x)=0\}.$$ Let us ...
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### A injective differential-form

Let $M$ be a smooth differential manifold. Let ω be a $1$-form on $M$ and $X$ be a fixed vector field on $M$. Then, in order to say "If $ω(X) = 0$, then $X = 0$ (i.e. $ω$ is injective?)", what ...
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### Are submanifolds of $\mathbb{R}^n$ cut out by smooth functions?

Let $M$ be a submanifold of $\mathbb{R}^n$. I was wondering can one always find (perhaps finite) smooth functions that cut out $M$? I was wondering about analogies between manifolds and (affine) ...
Suppose $f:(0,1)^n\to \mathbb{R}$ is continuous. Does the boundary of the set of its roots have Lebesgue measure 0? I guess the answer is negative, in that case, are there any reasonable conditions ...