# 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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### Is a map which sends a $3\times 3$ symmetric tensor to an element of $SO(3)$ which diagonalizes it necessarily discontinuous?

For a $3\times 3$ symmetric matrix $Q$, one can construct a map to $SO(3)$ which sends $Q$ to a matrix which diagonalizes it. If $Q$ has distinct eigenvalues, there are three choices for rotation ...
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### Asymptotic expansion of non analytical function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function. Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders ...
1 vote
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### Why does a function has to be differentiable so many times to be considered smooth?

I'm studying "Smoothness". If a function is once differentiable for all x's, shouldn't it be considered smooth? Because it does "look smooth" for all f(x), there's no way it will ...
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### Smooth function vs $L$-smooth function

Smooth function is a function that has derivatives in all domains, and $L$-smooth function is a function that satisfies $||\nabla f(x) - \nabla f(y)||_2 \leq L||x-y||_2$ for all $x,y$. What is the ...
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### For a smooth integral curve, $c$, if $\lim_{t \rightarrow \inf} c(t) = p$ does $\lim_{t \rightarrow \inf} c'(t) = 0$

If a smooth integral curve has a limit point $p$ and it takes infinite time to reach that point. Does that imply that its velocity approaches $0$ at that point? Or in other words, Updated based on ...
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### Why do we need tangent vector unequal to zero for smoothness of a vector function?

]1 My textbook gives this definition of smoothness of a $\vec r(t)$ on an interval $I$ of $t$. Why do we need $\vec r'(t) \neq\vec0$ on $I$?
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### differentials of smoothly varying family of maps

Let $F:N\times M \to M'$ be a smooth map, which we interpret as a "smooth" family of maps $M \to M'$, parametrized by N, so we have a map $F(y,\cdot):M \to M' \: \forall y \in N$. Show that ...
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### Construction of flat functions

The popular function f(x) = e^(-1/x^2) when x is nonzero = 0 when x = 0 defines one that is flat at 0 (all derivatives vanish there). My question is: How do I construct a (differentiable) ...
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### Construction of a immersion

It's possible construct a $C^\infty$ immersion $f$ of an open interval as in the next picture? Where the image curve has a part contained in the y-axis betwen -1 and 1, and self-intersections in the ...
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### How to prove product and quotient of smooth functions is smooth

I'm trying to prove the following problem: Let $A\subset\mathbb{R}^n$ be open. If $f, g: A \rightarrow \mathbb{R}$ are smooth, show that $fg$ and $f/g$ is smooth. (For the quotient case, $g$ is ...
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### Smooth Atlas of a Smooth Manifold

In Lee's Introduction to Smooth Manifold (2nd Ed.) the Proposition 2.5 gives two equivalent characterizations of smoothness of a map $F \colon M \to N$ between smooth manifolds $M$ and $N$, the second ...
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### Mathematical Definition of "Smoothness" : Requirement of Differentiable Functions to be Smooth?

I have often heard the following arguments made about the requirement for differentiable functions to be smooth: Fractal Functions are nowhere smooth Differentiable functions need to be smooth ...
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### For a Lie-group G and embedded Lie-subgroups K < H < G, prove that $\pi:G/K \rightarrow G/H$ is a surjective smooth submersion

The question is already in the title. For a Lie-group G and embedded Lie-subgroups $K < H < G$, prove that $\pi:G/K \rightarrow G/H$ is a surjective smooth submersion. Where it is meant that $K$ ...
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### Cartesian product of smooth functions is smooth.

Let $f_i: U_i \subseteq \mathbb{R}^{m_i}\to \mathbb{R}^{n_i}$ be functions that are smooth at $p_i \in U_i$, where $U_i$ is an open subset of $\mathbb{R}^{m_i}$, for $i=1,2$. I want to show that the ...
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### What is the domain of $g^* \circ f^*$?

Here is the question I am trying to prove: If $M$ is a smooth manifold, let $C^{\infty}(M)$ be the set of smooth functions $M \to \mathbb R.$ The set $C^{\infty}(M)$ is an $\mathbb R$-algebra under ...
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### Regarding Theorem 1.9, Conway, complex Analysis

I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable. I know that the estimation of the integral in equation 1.10, comes from the previous ...
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### Is the mapping from an empty set to an empty set smooth?

For a function/mapping of the form $f(x) = x^2$, we can perform differention to check if it is smooth. But for a mapping from empty set to empty set, what is the corresponding function $f(x)$ and how ...
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Given a sequence $(y_n)_n$ of real numbers, can we find a smooth ($C^1$, $C^2$ or even $C^\infty$) real function $\phi$ such that $\phi(2^n)=y_n$ for all $n\in\mathbb{N}$ ? It's clear if we want a ...
### Is $C^\infty_0(\mathbb{R}\!\setminus\!\{0\})$ dense in $L^2(\mathbb{R})$ with respect to the $L^2$-norm?
Why should that matter? I'd like to prove whether or not the operator $X^{-n}:D({X}^{-n})\subset L^2(\mathbb{R})\to L^2(\mathbb{R})$, with $n\in\mathbb{N}$, defined as $X^{-n} :f\mapsto{x^{-n}}f$ ...