# 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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### How to show $C^\infty_0$ dense in $C_0^1$ with respect to $W^{1,p}$-norm

I want to prove that $C^\infty_0$ is dense in $C_0^1$ with respect to $\|f\|^p=\|f\|_{L^p}^p+\|\nabla f\|_{L^p}^p$ or a reference for that proof. Or alternatively is $C_0^1$ contained in $W_0^{1,p}$, ...
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### What does it mean for an inclusion map to be smooth?

Sorry for the elementary question, but I do not understand what does it mean for an inclusion map to be "smooth"? My understanding is that an inclusion map will send an element $x$ into ...
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### Smooth functions with vanished derivatives at the boundary

I need to find a function $f$ in $[a,b]$ which satisfy that $$f(a) = f_1, \\ f(b) = f_2, \\ f^{(j)}(a) = 0, \quad j = 1,...,k, \\ f^{(j)}(b) = 0, \quad j = 1,...,k,$$ where $f^{(j)}$ is the j-th ...
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### Smooth function times continuous is Lipschitz?

Is a smooth compactly supported function multiplied by a continuous but not necessarily differentiable function already lipschitz continuous? right now Im just interested in the answer. If you know ...
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### Parameterization of area between $y=x+\cos x$ and $y=x+\sin x$

In the $xy$-plane a function is given: $f(x,y)=x+y$. Let $A$ be the area that is in the $xy$-plane and is encapsulated by $x=0$, $x=\frac{\pi}{4}$, $y=x+\cos(x)$ and $y=x+\sin(x)$. a) Make a ...
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### Which function space on $\mathbb{R}^n$ is identified with $C^\infty(S^n)$?

Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be $n$-sphere. Then, it is well-known that $S^n$ is the one-point compactification of $\mathbb{R}^n$. Now consider $C^\infty(S^n)$, the ...
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### On the smooth functions and differentiable manifolds [duplicate]

I'm new on stackexchange community (this is my second question on these topics) and this one is linked to my first (About differentiability of economics functions). Indeed, I asked on the economic ...
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### Is it really true that $\mathcal{S}(\mathbb{R}^n)$ is identified with smooth functions on $S^n$ vanishing at a fixed point?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space on $\mathbb{R}^n$ and $C^\infty(S^n)$ be the space of smooth functions on $n$-sphere. Now fix a point $x \in S^n$ and define C^\...
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### Density of $C_c^\infty(\Omega)$ in $L^p(\Omega)$ for a bounded open set $\Omega$ - any detailed proof?

Let $\Omega$ be an open, bounded domain in $\mathbb{R}^n$ (NOT necesasrily with smooth boundary). Now, let $C_c^\infty(\Omega)$ be the space of smooth functions on $\Omega$ with compact support. Then ...
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### Can a smooth curve have a segment of straight line?

Setting: we are given a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^n$ Informal Question: Is it possible that $\gamma$ is a straight line on $[a,b]$, but not a straight line on $[a,b]^c$? ...
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### What structure do smooth maps and diffeomorphisms preserve?

Context: Continuous maps between topological spaces are structure preserving in the following sense: Given two topological spaces $(X,\tau_X),(Y,\tau_Y)$ (where $(X,\tau_X)$ is the topological space ...
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### Can the construction of this 2D Curve of Constant Width be adapted to a 3D Surface of Constant Width?

A Surface of Constant Width is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the ...
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### I wonder why the author wrote $\phi_i:\mathbb{R}^n\to\mathbb{R}$ instead of $\phi_i:A\to\mathbb{R}$. ("Analysis on Manifolds" by James R. Munkres.)

I am reading "Analysis on Manifolds" by James R. Munkres. Theorem 16.3 (Existence of a partition of unity). Let $\mathcal{A}$ be a collection of open sets in $\mathbb{R}^n$; let $A$ be ...
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### Prove that if every smooth function on a subset of a manifold can be extended to a smooth function on the whole manifold, then the subset is closed.

Consider a nonempty subset $A$ of a smooth manifold $M$ and suppose that every smooth function on $A$ can be extended to a smooth function on $M$. We want to show that $A$ is closed. How might we ...
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### How do we know that there is a smooth function that is constant on two disjoint non-empty compact sets in $\mathbb{R}^n$?

Let $A$ and $B$ be non-empty compact sets in $\mathbb{R}^n$ such that $A\cap B\neq\emptyset.$ Prove that there is a smooth function $f$ on $\mathbb{R}^n$ with values ranging (inclusively) between 0 ...
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### Example of smooth compactly supported function

Is there an easy example of a smooth compactly supported function (ie. a test function) that equals $e^x$ on the interval $[-1,1]$? This is in reference to the following stack exchange question here, ...
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### Let $F: \mathbb R^n \to \mathbb R^m$ be smooth and such that for all $a \in \mathbb R$, $x \in \mathbb R^n$, $F(ax) = aF(x)$. Prove $F$ is linear

Let $F: \mathbb R^n \to \mathbb R^m$ be smooth and such that for all $a \in \mathbb R$, $x \in \mathbb R^n$, $F(ax) = aF(x)$. Prove $F$ is linear. I am sorry to say this, but I am really stuck on ...
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This paper defines the Restricted Strong Smoothness as follows: A differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be Restricted Strongly Smooth (RSS) with modulus $L_s>0$ or is $... • 175 1 vote 0 answers 72 views ### Smoothness of Fourier transform I am trying to understand, because the Fourier transform of the function$f(x) = e^{ -\sqrt{ \lvert x \rvert } }$is smooth. My question: Under which conditions is the Fourier transform of an$L^1$or ... 0 votes 1 answer 50 views ### Can smooth approximations always preserve injectivity? Most generaly, does every continuous injective mapping$f:M\rightarrow N$(these are smooth manifolds) have a smooth and regular injective approximation of arbitrary precision, given that$\dim(N)>\...
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I am new on computing external derivative, wedge product and pull-backs so I am having issues to understund some things about those things. For example, an excercise of my class notes is to prove that ...
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### I want you to check if I understand the case 1 in the proof of Theorem 3-11. (Partitions of unity in "Calculus on Manifolds" by Michael Spivak)

I am reading "Calculus on Manifolds" by Michael Spivak. I am now reading the case 1 in the proof of Theorem 3-11 about partitions of unity. Unfortunately I could not understand the case 1 in ...
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### What does "by (2) for each $x$ this sum is finite in some open set containing $x$" mean? ("Calculus on Manifolds" by Michael Spivak.)

3-11 Theorem. Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then there is a collection $\Phi$ of $C^\infty$ functions $\varphi$ defined in an open set containing $A$, with the ...
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### When is $f(z)$ analytic in the neigbourhood of $z$ for some complex $z$ close to the real line?

Let $f(x)$ be real-valued, 1-periodic, bounded and $C^{\infty}$ for real $x$. When is $f(z)$ analytic in the neigbourhood of $z$ for some complex $z$ close to the real line ? Notice this is similar to ...
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