Mathematics

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.

877 questions
65
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9 answers

Are there other kinds of bump functions than $e^\frac1{x^2-1}$?

I've only seen the bump function $e^\frac1{x^2-1}$ so far. Where could I find examples of functions $C^∞$ on $\mathbb{R}$ that are zero everywhere except on $(-1,1)$? Are there others that do not involve the exponential function? Are there any with…
jnm2
  • 3,260
33
votes
10 answers

Formula for bump function

I would like to formulate a bump function ( link ) $f:\Bbb R \to\Bbb R$ with the following properties on the reals: $$ f(x) = \begin{cases} 0, & \mbox{if } x \le -1 \\ 1, & \mbox{if } x = 0 \\ 0, & \mbox{if } x \ge 1 \end{cases} $$ In addition…
Richard Burke
  • 945
29
votes
1 answer

Is there any simple set of properties which uniquely characterizes differentiation?

The transformation of differentiation is a linear operator over $C^\infty(\mathbb{R}),$ the vector space of smooth functions over $\mathbb{R}.$ Is there any simple set of properties that uniquely determines this linear operator other than the…
mathlander
  • 4,313
21
votes
2 answers

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

Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}$ ? Note: Here the answer is NOT required to be an elementary function, as I already know that otherwise the answer would…
blademan9999
  • 814
18
votes
3 answers

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 $[a,b]^c$ ? Formal Question: It is possible that…
John Frank
  • 962
18
votes
3 answers

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 said point. The thread is closed with an accepted…
jeff honky
  • 886
17
votes
3 answers

vector space of all smooth functions has infinite dimension

Now, I am working through a particular case in the book on smooth manifolds by John.M.Lee used in my graduate math class, let's say we have a smooth manifold X which has positive dimension. He then claims that the vector space $C^\infty$ of all…
user8169
16
votes
2 answers

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 domain is a smooth manifold with boundary. Lee himself…
Hempelicious
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14
votes
4 answers

Is there an intrinsic approach to defining manifolds?

I don't know much about the fondations of the theory of manifolds, but the way additional structure if defined on manifolds doesn't feel right to me. For example, to define a smooth manifold, we first define what it means for maps in $\mathbb{R}^n$…
Carl Chaanin
  • 733
14
votes
1 answer

Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind)

Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind) In this other question I did about Bump functions , other user told in an answer that these kind of functions "tends aggressively fast to zero at the limits of the…
Joako
  • 2,237
14
votes
2 answers

Particularly nice bump function

Is there a smooth ( $C^\infty$ ) function $f: \mathbb{R} \to \left[ 0, 1 \right]$ such that: $f(x) = 1$ iff $x = 0$ , $f(x) = 0$ iff $\left\lvert x \right\rvert \geq 1$ , and $0 < f(x) < 1$ otherwise; Every derivative of $f$ is simultaneously $0$ at…
Mel
  • 306
11
votes
5 answers

If $f(x)$ is smooth and odd, must $f(x)/x$ be smooth?

Let $f:\mathbb R\to\mathbb R$ be: smooth, i.e., infinitely differentiable, and odd, i.e., $f(x)=-f(-x)$ for all $x$ . Let $g:\mathbb R\to\mathbb R$ be defined as $$g(x):=\left\{ \matrix{f(x)/x, & x\neq 0 \\ \lim_{x\to 0} f(x)/x, & x=0} \right.$$…
WillG
  • 7,762
10
votes
1 answer

Does little Bézout theorem hold for smooth functions?

As a special case of little Bézout theorem, if we have a polynomail $f(x)$ with $f(0)=0$, then there exists another polynomial $g(x)$ such that $f(x)=xg(x)$. It's easy to see that this fact generalizes to analytic functions because we have Taylor…
Syang Chen
  • 3,576
10
votes
3 answers

Is the following piecewise-defined function smooth $\in C^\infty(\mathbb{R})$?

Is the function $$q(x)=\begin{cases} 1& x=0\\ 0& |x|\geq 1\\ \dfrac{1}{1+\exp\left(\dfrac{1-2|x|}{x^2-|x|}\right)}&\text{otherwise}\end{cases}$$ a smooth function class $C^\infty(\mathbb{R})$ ? I found this function as is shown in Wiki by choosing…
Joako
  • 2,237
10
votes
3 answers

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 have sharp corners or cusps because it's differentiable. Then why does…
Orange Cat
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