# Tag Info

• 112k
Accepted

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

$\textbf{Hint: }$ After you choose vectors $v_{k+1},\dots,v_n$, you have linearly independent vectors on $T_pM$. Extend $\{v_{k+1},\dots,v_n\}$ around a neighbourhood of $p$, say to constant local ...
• 7,000
Accepted

### Is there a simple way to characterize the smooth functions without using the derivative?

Consider data $(V,D)$ with the following properties: $$\tag1 V\text{ is a subalgebra of }\Bbb R^{\Bbb R}$$ $$\tag2 (\mathbf 1\colon x\mapsto1)\in V$$ $$\tag3 (\operatorname{id}\colon x\mapsto x)\in V$$...
• 374k
Accepted

### Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}...$

The link-only comment of metamorphy is actually a full answer, and gives an analytic function, instead of the smooth function of Michael's answer. Instead of hiding behind a link to Wikipedia, I give ...
• 34.8k
Accepted

### Can a smooth map be extended to an open set?

Yes, and the construction is by means of a partition of unity. That is, consider a cover of $X$ by open sets $U_\alpha$, and for each $\alpha$, find a function $g_\alpha:U_\alpha\to\mathbb R^m$ such ...
• 24.5k
Accepted

### Does there exist smooth functions $f_i,g_i \in C^{\infty} (\mathbb R)$ such that $\sin (xy) = \sum\limits_{i = 1}^{n}f_i (x) g_i (y)$ for all $x,y\$?

Taking $y = 1,..., n+1$ implies that $\sin x, \sin 2x, ... , \sin((n+1)x)$ are a subset an $n$-dimensional space of functions. This will contradict that these functions are linearly independent.
• 44.8k
Accepted

### Can a non-zero smooth $f: [a,b] \rightarrow \mathbb{R}_{\geq 0}$ have infinitely many zeroes?

Let's start with $f_1(x) = \sin(1/x)$, which has infinitely many zeroes, but it's clearly not smooth (not even continuous). You can square it to get a nonnegative function $f_2(x) = \sin^2(1/x)$. As ...
• 5,921
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### Why are coordinate maps diffeomorphisms? (A seeming counterexample?)

You have two different copies of $\Bbb{R}$ and they’re playing different roles, but you’re not distinguishing them. The first, lets call it $M$ to emphasize it is the ‘abstract manifold’ has a global ...
• 54.2k

• 1,328
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### Connecting smooth functions in a smooth way

There is a well known result of Borel that will be useful here. Borel's theorem on power series: Let $x_0\in \mathbb R.$ Given $a_0,a_1, \dots \in \mathbb R,$ there exists $f\in C^\infty(\mathbb R)$ ...
• 105k
Accepted

### Can a function be smooth at a single point?

As I suggested yesterday, I will try to fix the example Paul Sinclair made in this post. Let $f_0$ be a continuous and nowhere differentiable function. We define recursively for $n\geq 0$ as follows: ...
• 18.6k

• 1,283
### Continuous function for day/night with night being $c$ times longer than day
In the notation of my "answer" to my still-unanswered question Almost simple Hermite interpolation, we can compute a quintic polynomial $l_a(x)$ such that $l_a(0) = l_a(1) = 0,$ $l_a(a) = 1,$...