# 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.

156 questions
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### 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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### 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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### 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 $f(n)=g(w)$. ...
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### “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 ...
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### 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 ...
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### 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, ...
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### 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-...
### 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 ...