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.

11
votes
3answers
294 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 ...
6
votes
2answers
87 views

Does an integral inequality imply a pointwise inequality?

$\newcommand{\R}{\mathbb{R}}$ Let $(f_n)_n \subset L^1(\R^N)$. Suppose that for any nonnegative function $\phi \in C_c^{\infty}(\R^N)$, we have: $$ 0 \leq \liminf_{n \rightarrow \infty} \int f_n \, \...
6
votes
0answers
79 views

The identity theorem at the boundary (complex analysis)

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{C}$ be a smooth map, which is holomorphic on the open unit disk $\text{int}(\mathbb{D}^2)$. Suppose that there exists ...
6
votes
0answers
71 views

A Smooth Function with Peculiar Properties

Suppose $f:\mathbb R^n\to\mathbb R$ is a smooth function and define $E_f := \{x\in\mathbb R^n\;|\;f(x)=0\text{ and }\nabla f(x)\ne 0\}$. Can we find $f$ such that $E_f$ has positive $n$-dimensional ...
4
votes
1answer
202 views

Completion of local frames for the tangent bundle of a smooth manifold

In John M. Lee book Introductions to smooth manifolds Proposition 8.11 is left as exercise. Can anyone give me hints and suggestions to prove it? In particular, i want to show: Let $M$ be a smooth ...
4
votes
2answers
172 views

Connecting smooth functions in a smooth way

Let $a<b<c<d$ be real values, and let $f \in C^{\infty}([a,b])$ and $g \in C^{\infty}([c,d])$. Is there a way to "connect" these functions in a smooth way? That is, is there a function $h \in ...
4
votes
1answer
262 views

Discrepancy between calculus methodologies - Is it significant?

Two of the ways of doing calculus with algebra are non-standard analysis NSA and smooth infinitesimal analysis SIA. NSA has a technique called 'taking the standard part' which neglects incremental (or ...
3
votes
2answers
48 views

Straight Lines are Strict Minimizers of Arclength in Euclidean Space

Let $a$ and $b$ be two points in an euclidean space $\mathbb{R^n}$ Then,I denote by $\Omega(a,b)$ set of all maps $\gamma : [0,1] \to \mathbb{R}^n$ such that $\gamma(0) = a,\gamma(1) = b$ and the ...
3
votes
2answers
68 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 ...
3
votes
1answer
39 views

Is the integral of a smooth function on a closed smooth real manifold finite?

Let $M$ be a closed smooth real manifold and $f$ a $C^\infty(M)$ function. I need to prove whether $$\int_M f(p)\mathbb dp\in\Bbb R$$ I think yes, because for all $p\in M$, $f(p)\leq k$ for some $k\...
3
votes
2answers
118 views

Questions on the real bump function and conjuction of smooth functions

Before I ask you my question which I will mark in bold I will tell you what I already gathered so far. In a previous result I have showed that the bumpfunction is smooth. The bumpfunction is defined ...
3
votes
1answer
150 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 ...
3
votes
2answers
147 views

If a composition of functions is smooth and one of them is smooth, then the other is smooth

Show that a map $\xi$ between smooth manifolds $M$ and $N$ is smooth if and only if $f ◦\xi$ is a smooth function on $M$ whenever $f$ is a smooth function on $N$. One implication is clear because I ...
3
votes
2answers
221 views

Smooth images of manifolds are immersed?

in various papers in symplectic geometry, I have encountered the following argument. Statement: Suppose $f: M \rightarrow N$ is a smooth map of constant rank. Then its image $f(M)$ can be equipped ...
3
votes
0answers
50 views

Confusion on $\vert\vert \varphi\vert\vert_{p}$ where $p \in [1,\infty[$

Let $\varphi \in C^{\infty}(\mathbb R^n)$ and for $\epsilon > 0$ define $\varphi_{\epsilon}(x):=\epsilon^{-n}\varphi(x/\epsilon)$ such that $\varphi_{\epsilon} \in C^{\infty}(\mathbb R^n)$ with ...
3
votes
0answers
55 views

If $C\subset\mathbb{R}^n$ is closed, then $C=f^{-1}(0)$ for some smooth $f:\mathbb{R}^n\to\mathbb{R}$.

Let $C\subset \mathbb{R}^n$ closed. Prove that there is a smooth function $f:\mathbb{R}^n\to\mathbb{R}$ such that $C=f^{-1}(0)$. I've found this solution in the internet: take a cover of balls $\{B_i\...
3
votes
1answer
111 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\|...
3
votes
0answers
231 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 ...
3
votes
2answers
79 views

Intuition of a Smooth Curve

I'm currently taking an intro to Differential Geometry course, and am having trouble with the definition of a smooth curve. If you consider the curve $\lambda(t) = \left (\cos^3(t),\sin^3(t)\right )$ ...
2
votes
2answers
60 views

How do I show that this set is dense in the function space?

Consider the smooth manifold $\mathbb{S}^3$ embedded in $\mathbb{R}^4$, note that $$\widetilde{T}:= \frac{1}{\sqrt{2}}\mathbb{T}^2 = \left\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4; \ x_1^2 + x_2^2 = x_3^2 ...
2
votes
1answer
46 views

Is $C_{c}^{\infty}[a,b]$ a complete metric space?

Is the space of smooth functions with compact support in an interval $[a,b] \subset \mathbb{R}$, with the metric $\rho(\varphi_{1},\varphi_{2})= \sum_{n=0}^{\infty}2^{-n}\frac{\left \| \varphi_{1} - \...
2
votes
1answer
66 views

Is any $C_c^\infty(\mathbb R)$-function the product of two $C_c^\infty(\mathbb R)$-functions?

Let $f\in C_c^\infty(\mathbb R)$. Are we able to write $$f=gh$$ for some $g,h\in C_c^\infty(\mathbb R)$? Unfortunately, I've no idea how to I could prove or disprove this.
2
votes
1answer
76 views

$[a,b]$ as a smooth manifold with boundary has global coordinates?

Consider a compact interval $[a,b]$. If $[a,b]$ had global coordinates, then there would be an homeomorphism $f:[a,b]\to U$ where $U$ is an open subset of $\mathbb{R}$ or an open subset of $[0,\infty)$...
2
votes
1answer
41 views

Smooth curve segments and smooth charts

Let $M$ be a smooth manifold with or without boundary, and let $\gamma:[a,b]\to M$ be a smooth curve. I want to show that there exist a finite partition $a=a_0<a_1<\dots<a_k=b$ such that $\...
2
votes
2answers
40 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 ...
2
votes
1answer
54 views

smoothness of a map that comes from a smooth submersion

Let $f:N\rightarrow M$ be a surjective submersion. Let $g:W\rightarrow M$ be a smooth map. Let $w\in W$. We have $g(w)\in M$. As $f$ is surjective, there exists $n\in N$ such that $f(n)=g(w)$. ...
2
votes
2answers
94 views

“Standard reference” for $C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$

$C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$. This can be shown by mollification. This is a well-known, widely used fact. However, I wasn't able to find any book which I could point in a ...
2
votes
2answers
66 views

Clarification on definition of smooth map between smooth manifolds

Let $M$ and $N$ be $n$-dimensional smooth manifolds. A map $F: M \to N$ is smooth if for each $p \in M$ there exists smooth charts $(U, \varphi)$ containing $p$ and $(V, \psi)$ containing $F(p)$ such ...
2
votes
1answer
66 views

How to show a real valued function of several variables is analytic?

Let $f:\Omega\subset\Bbb{R}^m\to\Bbb{R}^n$ be a given function. In general, how can I show that $f$ is smooth (infinitely differentiable) and analytic. I know this is a bit vaguely stated question, ...
2
votes
1answer
28 views

Injective smooth map which is immersive and submersive is actually a global diffeomorphism

Let $U\subset \Bbb R^n$ be an open subset and $f:U\to \Bbb R^n$ be an injective smooth map where $Df_p$ is invertible for each $p\in U$. I want to show that $f$ is a global diffeomorphism from $U$ to $...
2
votes
1answer
49 views

Does an immersed curve have to be “curve-like” somewhere?

Let $C$ be a smooth immersed curve, i.e. an image of a smooth immersion $\varphi\colon \mathbb{R} \to \mathbb{R}^n$. We will say that $C$ is curve-like at a point $p \in C$ if, for every two smooth ...
2
votes
1answer
39 views

How can I create a cubic spline to connect these two line segments?

I am trying to create an audio envelope smoothing curve to smooth a steady signal (y=1) as it goes into exponential decay (y=c^-x). I have expressed the graphs of the two line segments I am trying to ...
2
votes
1answer
63 views

Smooth, approximately space-filling curves in high dimensions

I'm looking for smooth (infinitely differentiable everywhere) functions (curves) $\mathbb{R}\rightarrow\mathbb{R}^d$ that are approximately space-filling, i.e. scaling allows the curve to eventually ...
2
votes
1answer
75 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 ...
2
votes
1answer
33 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 ...
2
votes
1answer
88 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 ...
2
votes
2answers
107 views

Are charts for smooth manifolds homeomorphisms or diffeomorphisms?

I will link the following lecture notes, because it makes no sense to keep pasting from them. When reading them, there are two things I do not understand. The author introduces smooth manifolds by ...
2
votes
0answers
38 views

Transformation between two measures

If $\mu$ and $\nu$ are two measures, both absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^d$ with smooth densities $p_\mu(\mathbb{x})$ and $p_\nu(\mathbf{x})$, does it always ...
2
votes
0answers
30 views

Perimeter of level sets of a smooth function

I've a simple question concerning the perimeter of level sets of a smooth function. Let $f:\Omega \to \mathbb{R}$ be a smooth function defined on a bounded domain of $\mathbb{R}^n$. We set $A_s:=\{f&...
2
votes
1answer
32 views

Smooth functions with equal derivatiesof all orders in one point

Let's assume that $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are smooth functions and $\forall n\in \mathbb{N}, \forall x\in \mathbb{R}:\, |f^{(n)}(x)|, |g^{(n)}(x)| < 1 $ prove that if $\forall n ...
2
votes
0answers
58 views

About change of coordinates of vector fields in smooth manifold theory

Background Suppose $M$ is a smooth $n$-manifold and $(U,\varphi=(x^i))$ a smooth chart on $M$. For each point $p\in U$ we know that $(\frac{\partial}{\partial x^i}|_p)_{i=1}^n$ is a basis for the ...
2
votes
1answer
67 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-&...
2
votes
0answers
50 views

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

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 $\...
2
votes
0answers
79 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 ...
2
votes
0answers
46 views

Show that rational function of $C^k$ polynomials is $C^k$

Let $E \subseteq \mathbb{R}^n$ be an open set. Suppose $$P,Q: E \to \mathbb{R}$$ are polynomial functions in several variables. For example $$P(x,y) = x^2y + x + y^2 +1$$ Suppose $Q$ ...
2
votes
1answer
73 views

Is it possible $\cos \phi (s)$ and $\sin \phi (s)$ to be nth order differentiable and not $\phi (s)$?

Pressley's Differential Geometry book supposes that the angle between the tangent vector to a curve and any fixed vector is smooth (i.e. all orders of differentiablity exits and are continuous) and in ...
2
votes
0answers
43 views

For what functions is $\lim_{n\to \infty}|f^{(n)}(x)|=0$? (Where $f^{(n)}(x)$ is the $n$th derivative of $f$)

I recently came across a formula that involved the assumption that $\lim_{n\to \infty}|f^{(n)}(x)|=0$, and at first I thought that it would be true for every function, but then I remembered that ...
2
votes
1answer
36 views

Find a sequence $(\phi_n)_n \subset C^{\infty}_c(\mathbb{R}^N)$ which converges in both $L^p(\nu)$ and $L^q(\mu)$ to $1_E$

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Cr}{C^{\infty}_c(\R^N)}$ Suppose we have two non-zero Borel measures on $\R^N$, labeled $\nu$ and $\mu$, and we have $1 \leq p, q < \infty$. Let $E \...
1
vote
2answers
42 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-...
1
vote
3answers
88 views

if $d_f \neq 0$ then f is a submersion

I often see in my course of differential geometry that $d_f$ the differential of a smooth function $f$ is sumersive iff $d_f \neq 0$. Is this a true statement, if yes, how could we prove that? I ...