Linked Questions

4
votes
1answer
706 views

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 ...
0
votes
0answers
59 views

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 ...
13
votes
4answers
406 views

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: ...
3
votes
4answers
301 views

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 ...
4
votes
1answer
120 views

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 ...
4
votes
1answer
149 views

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 $f=0$?
1
vote
1answer
60 views

Zero locus of a set of smooth functions. Smooth analogue of affine varieties.

The notion of an affine variety is one of the most fundamental concepts in algebraic geometry. It is just the zero locus of some finite set of polynomials $f_1,\dots,f_k$ from $\mathbb{F}[X_1,\dots,...
1
vote
1answer
72 views

How smooth is the boundary of a set defined by a function with certain smoothness?

Given a non-negative function $\phi\colon\mathbb R^n\to\mathbb R$, which is in $C^k$, and define the set \begin{align} S:=\{x\in\mathbb R^k\colon \phi(x)>0\} \end{align} then can we say that $\...
3
votes
1answer
88 views

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 ...
2
votes
1answer
71 views

$V$ 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 $V=\{x\in \mathbb R:f(x)>0\}$
0
votes
1answer
39 views

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 ...
0
votes
1answer
38 views

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 ...
2
votes
0answers
46 views

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) ...
2
votes
1answer
35 views

Lebesgue measure of boundary of sets of roots

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 ...
1
vote
1answer
37 views

zeroes of convergent fourier series

Recently I learned that the distribution of zeroes of a continuous real-valued function in a closed interval can be such that they might be uncountable. An example was given by Daniel Fischer who ...