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Questions tagged [smooth-functions]

For questions about infinitely or arbitrarily differentiable (smooth) functions of one or several variables. To be used especially for real-valued functions; for complex-valued functions, the tag holomorphic-functions is more appropriate.

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7 views

Smooth endpoint map

Consider a control system $$ \dot x(t) =f(x(t),u(t)) \qquad (\star) $$ where $f$ is a smooth vector field and $x\in \mathbb{R}^n$. The endpoint mapping is defined by $$ E:\mathbb{R}^n\times \mathbb{...
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1answer
43 views

Smooth map with null differential at each point are constant on the connected component of the domain

Let $F:M\to N$ be a smooth map between smooth manifolds $M$ and $N$ (with or without boundary). I want to show that $dF_p:T_pM\to T_{F(p)}N$ is the zero map for each $p\in M$ if and only if $F$ is ...
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1answer
46 views

Definition of smooth functions on arbitrary subsets of $\mathbb{R}^n$ and partial derivatives

Let $A$ be an arbitrary subset of $\mathbb{R}^n$, and let $f:A\to \mathbb{R}$ be a function. We say that $f$ is smooth if for each point $p$ in $A$ there exists an open subset $U$ of $\mathbb{R}^n$ ...
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2answers
32 views

Smooth curves and velocity

Let $M$ be a smooth manifold. Let $X:M \to TM$ be a global smooth vector field on $M$, and let $K$ be the support of $X$, i.e. $K=\overline{\{p\in M: X_p\not=0\}}$. Suppose $K$ a is compact subset of ...
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3answers
253 views

Can a function be smooth at a single point?

I saw a thread (Find a function smooth at one isolated point) in which it is asked whether or not it is possible for a function to be smooth at a point, but not smooth on a deleted neighbourhood of ...
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1answer
75 views

Smooth Elementary Function that Outgrows All Tower Functions?

This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer. Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth. Is ...
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1answer
56 views

Computing the differential of a certain smooth map

Let $M \subseteq \mathbb{R}^k$ be an embedded submanifold of $\mathbb{R}^k$, with dim$M=n$. Let $v$ be in $\mathbb{S}^{k-1}$, and let $P_v:\mathbb{R}^k\to(\mathbb{R}v)^{\bot}$ defined by $P_v(x)=x-&...
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1answer
57 views

Submanifolds and adapted atlas

Let $M$ be a smooth manifold of dimension $n$. My notes say Theorem: A subset $S$ of $M$ could be given a structure of smooth manifold of dimension $k$ such that $S$ is an embedded submanifold ...
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1answer
43 views

Adapted charts for smooth manifold

Let $M$ be a smooth manifold of dimension $n$ and let $S$ be an embedded submanifold of $M$, i.e. a subset of $M$ which is given a structure of smooth manifold of dimension $k\leq n$ such that the ...
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43 views

Theorem about smooth functions on manifolds

Here is a theorem as it appears in Spivak's Differential Geometry Vol. 1. THEOREM. (1) If $f:M^n\to N^m$ has rank $k$ at $p$, then there is some coordinate system $(x,U)$ around $p$ and some ...
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1answer
108 views

Sequences of bounded smooth functions versus numerical sequences of their supremum norms

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of smooth functions $f_n:\mathbb{R}\rightarrow\mathbb{R}$ which are bounded together with all their derivatives $f^{(j)}_n$, $j,n\in\mathbb{N}$, and $(\|f_n\|...
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1answer
30 views

Proving that the image of an injective, proper immersion is a manifold

I am trying to get through the proof of the statement "if $f: M \to N$ is injective, proper and an immersion, then $f:M \to f(M)$ is a diffeomorphism onto a submanifold". The proof I'm reading says ...
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0answers
9 views

Continuous on a disk, smooth on a disk minus a closed interval. Does it extend?

Suppose $u:\mathbb{D}\to\mathbb{R}$ is a continuous function, where $\mathbb{D}$ is the open unit disk. Let $I$ be a closed segment in $\mathbb{D}$. To make things simpler we can suppose $I=[a,b]$ is ...
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1answer
19 views

Approximating Sobolev functions in $W^{1,p}(\mathbb{R}_+^n)$

Let $p \geq 1$. I know that there exists a continuous and linear extension operator $$ E : W^{1,p}(\mathbb{R}_+^n) \to W^{1,p}(\mathbb{R}^n) .$$ I read that from the existence of such an extension ...
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2answers
54 views

If $f:\mathbb{R}\to\mathbb{R}$ is infinitely-differentiable, and $f(x+y)-f(y-x)=2xf^\prime(y)$, then it is a polynomial of degree less than $2$

$S$ is set of family of infinite differentiable function from $\mathbb R \to \mathbb R$ with $\forall x,y\in R$ $$f(x+y)-f(y-x)=2xf^\prime(y)$$ then I have to prove that $S$ only contain all ...
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1answer
41 views

Bounding the extrema of polynomials from $\frac{d^n}{dx^n} \exp(-1/x)$

As laid out on Wikipedia, the function $$f(x):=\begin{cases} \exp(-1/x) & x>0\\ 0 & x\le 0 \end{cases}$$ has the expression for derivatives at $x>0$, $$ f^{(n)}(x) = \frac{p_n(x)}{x^{...
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2answers
39 views

Finding perpendicular unit vector in $\mathbb{R}^n$ to hyperplane

Suppose I have $n-1$ linearly independent vectors $(v_1, ..., v_{n-1})$ in $\mathbb{R}^n$ that together form a basis of a hyperplane. I'm looking to find a last vector $v$ that is normal to the ...
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1answer
39 views

Proving Hopf's Fibration $\pi: \mathbb{S}^{2n+1}\to\mathbb{CP}^n$ is a submersion

Prove that the following map is a smooth, surjective submersion: \begin{align*} \pi:\mathbb{S}^{2n+1}&\to\mathbb{CP}^n\\ (x_0,y_0,...,x_n,y_n)&\mapsto [x_0+iy_0:...:x_n+iy_n] \end{align*} ...
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1answer
42 views

For a convex function $f$, is the following set convex: $X := \{x ∶ -f(x) \leq 1 \}$? [closed]

For a convex function $f$, is the following set convex: $X := \{x ∶ -f(x) \leq 1 \}$? I know that the set $X := \{x ∶ f(x) \leq 1\}$ is convex, but I'm unsure about the $-f(x)$ in the first set.
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1answer
49 views

Proof that β-function ∈ C^∞

I need to find correct proof that β-function is smooth on its domain. Is there some feature of such functions, I guess that we need to prove the continuity of all n-derivatives, or their partial ...
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42 views

Approximation of one function with smooth functions

There is a continuous but non-smooth function at $x=0$: $$ f(x)=\left[\frac{2}{1+e^{-2x}}-1 \right]_{+},$$ where $[u]_+\equiv \begin{cases} u,\quad u \geqslant 0\\ 0, \quad u<0 \end{cases} $ So, $...
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0answers
128 views

Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says: Formulate and prove a version of the rank theorem for a map of constant rank whose ...
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1answer
32 views

Is there a function for which the sum converges while the limit diverges?

Once I take my integral of f'(x) from n+1 to n I'll get f(n+1)-f(n). Since I already needed a function that's limit diverged, shouldn't my sum also diverge? This leads me to believe that this is a ...
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1answer
24 views

Existence of a $C_c^{\infty}$ function $\Phi$ s.t. $\int \nabla \Phi \neq 0.$

Let $\Omega\subseteq \mathbb R^n$ be open , bounded and connected set with $\mathcal{L^n}(\Omega)>0$. Let $\Omega=\Omega_1\cup\Omega_2$ where $\Omega_1\cap\Omega_2=\emptyset$ with $\mathcal{L^n}(\...
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1answer
40 views

Find linear transformation whose kernel is given

Question: given $V=C^∞(-∞,∞)$ i.e the vector space of real-valued continuous functions with continuous derivatives of all orders on $(-∞,∞)$ and $W=F(-∞,∞)$ the vector space of real-valued functions ...
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0answers
17 views

Constructing a uniform convergence sequence

Let $f : \overline{\Omega} \subset \mathbb{R}^{N} \to \mathbb{R}$ be a $C^{2}(\overline{\Omega})$ function. Can we always construct a sequence $f_{n}$ such that $f_{n} \to f$ uniformly in $\overline{\...
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1answer
25 views

How to check the smoothness of the following function?

I was solving a problem which boiled down to checking the smoothness of the following function. $$ \tilde{p} (z) = \begin{cases} \frac{1}{p(1/z)} &\quad z\neq 0 \\ ...
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0answers
30 views

Properties of Complex Function $f(z)=e^{-\frac{1}{z^{1/3}}}$

This post will be about a part of an example from my complex analysis book. Problem: They claim that there exist a function $J_+(z)$ holomorphic in the upper half plane $\operatorname{im}z>0$, ...
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28 views

Smooth extension of function on $\mathbb{R}^p$

I have the following specific problem: For an application I use the following function $g : \Bbb R^{Nd} \to [0,\infty)$, $$(x^1, \ldots, x^N) \mapsto \begin{cases} \prod\limits_{\substack{(a,b)\in\{...
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2answers
66 views

$\partial U$ is a smooth curve (i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}' \not= 0$)?

In a section discussing global maxima/minima, my textbook says the following: Simply stated, $\mathbf{x}_0$ and $\mathbf{x}_1$ are points where $f$ assumes its largest and smallest values. As ...
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0answers
16 views

Construct a $C^2$-continuous function for symmetric weighting

Let $\alpha>1$. I am looking for a $C^2$-continuous function $w:[0,\infty)\to\mathbb{R}$ which satisfies the following: \begin{align*} w(x)=1, &\ \ \ \ \ \ x\in[0,\frac{1}{\alpha}],\\ w(x)=1-w(...
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1answer
25 views

Proof that the product of locally integrable function $g$ with any test function is zero implies $g=0$ a.e

How to prove that If for $g\in L_{p}$ $p\in(1,2]$ and for all $\phi \in C_{0}^{\infty}$ we have $\int_{\mathbb{R}} g \phi d\mu =0 $ then $g=0$ a.e where $\mu $ is the Lebesgue measure. I was ...
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0answers
18 views

What conditions can the shape parameters satisfy to make a generalized Gaussian function a smooth function?

Generalized Gaussian function (GGF): $$g(x_j;\mu,\theta,q):=\frac{q}{2\theta\Gamma(1/q)}\exp\left\{-(\frac{|x_j-\mu|}{\theta})^q\right\},$$ where $\mu, \theta>0, q>0$ are mean, scale parameter ...
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0answers
8 views

Extending function with local maximum in a bounded smooth domain.

Let $\Omega \subset \mathbb{R}^{N}$ be a bounded $C^{2}$-domain. Take $V$ as a subset of $\Omega$ and define a smooth function $\phi$ such that $\phi \in C^{2}(\overline{V})$ and attains a local ...
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1answer
41 views

Decreasing smooth approximation of an indicator function with bounded derivative

Consider an indicator function $\chi_{[a,b]}$ on an interval $[a,b]$. My question - is there a decreasing sequence $f_\delta\in C^1(V)$ (where $[a,b]\subset V$ and V is open), such that $f_\delta\...
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1answer
34 views

Can I find an infinitely differentiable function of of bounded moments closest to triangle wave?

Based on this question regarding existance of closest function in Schwarz class, where answer was negative. What if we add a new constraint. Not only infinitely differentiable compact support but with ...
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2answers
31 views

Which function in the Schwarz class of functions is “closest” to triangle wave in $L^2$ sense?

Would it be possible to calculate which function in the Schwarz class of infinitely differentiable functions with compact support is closest to triangle wave? Let us measure closeness as $$<f-g,f-...
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1answer
39 views

How to show that $\exists$ $\phi\in C_c^{\infty}(\Omega)$ s.t. $\int_{\Omega_1}D\phi(x)dx$ has unit norm?

Let $\Omega\subset \mathbb R^n$ be open, connected and $\Omega=\Omega_1 \cup \Omega_2$ where $\Omega_1\cap \Omega_2=\emptyset$, $\mu(\Omega_1)>0,\mu(\Omega_2)>0.$ Then show that there exists $\...
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1answer
56 views

A more wide class than smooth functions

I will call a function $f$ almost smooth when for every point $a$ there are numbers $y_0$, $y_1$, $y_2$ ... $$f(x) = y_0 + \frac{y_1}{1!}(x-a) + \dots + \frac{y_n}{n!}(x-a) ^n + O((x-a)^{n+1})$$ for ...
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0answers
31 views

Prove that a function is smooth easily

In this answer I claimed that the function $f (x) = x^{n + 1} \sin \exp \frac{1}{x}$ is smooth in all points except zero. But how to prove this claim easily (without calculating the derivative ...
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1answer
66 views

$C^n$ function with big/small O notation

Can the statement that $f$ is a $C^n$ function (say for $\mathbb{R}\to\mathbb{R}$ functions) be written in terms of big/small O? Maybe it is equivalent to existence of numbers $f(a),\dots,f^{(n)}(a)$ ...
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1answer
56 views

Function that are not smooth because $f(U) \not \subset V$

In the book of Lee (page 35 of second edition) on smooth manifolds, for a map $f : M \rightarrow N$ between manifolds to be smooth, we have to prove in particular that for each point p in $M$ there is ...
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1answer
21 views

Sigmoid functions similar to atan but with different upper and lower bound

Do there exist smooth sigmoid functions similar to atan but with different upper and lower bounds? This function must go through origin. Something like on the picture: Function If not, why? Where can ...
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1answer
36 views

existence of a smooth function with bounded derivative

I am studying a proof in a differential geometry book where the author use a a smooth function satisfying some properties. Here is what it is claimed. $\forall 0 < \epsilon < \pi$ there exists a ...
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1answer
37 views

Smooth Urysohn function having $0$ as regular value

Let $M$ be a manifold and $C_0,C_1$ be two disjoint closed subsets of $M$ , then smooth Urysohn Lemma says that there exists a smooth function $f:M\rightarrow [0,1]$ such that $f(C_0)=\{0\},f(C_1)=\{1\...
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1answer
67 views

Smooth and not analytic

Can someone show me, without reference to Taylor series, why a complex function can be smooth but not analytic? I do not understand it intuitively or visually either. I would like an explanation ...
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1answer
45 views

Proof verification on the openness of the space of matrices of full rank

Let be $m<n$ and $M_m(m\times n, \mathbb{R})$ be the set of the matrices $m \times n$ of full rank $m$. I want to show that $M_m(m\times n,\mathbb{R})$ is an open subset of $M(m\times n,\mathbb{R}...
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1answer
95 views

Existence of a smooth map from the Circle to the Square

I know that there is no diffeomorphism from the unit circle, $S^1$ to the square of side length 2 centered at 0. However, can we construct a bijective map from $f : \mathbb{R}^2 \rightarrow \mathbb{R}...
2
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0answers
37 views

C^infty approximation of Heaviside step function: definite integral of a bump function

I'm wondering if there is an analytical form for a $C^\infty$ approximation with a compact support of a Heaviside step function $f(x) = I_{x \geq 0}$. In attempting to construct one, I'm taking a bump ...
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1answer
33 views

Existence of a smooth extension vanishes at infinity

in my research I saw this argument used in an article and I don't know if it's true (I believe yes). Let $\Omega \subset \mathbb{R}^3$ be an open, bounded and simply connected domain and let $\...