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.
Questions tagged [smooth-functions]
865 questions
65
votes
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…
jnm2
- 3,260
32
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
- 925
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
- 4,097
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 be…
blademan9999
- 796
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
- 857
17
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
- 846
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
15
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
- 295
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
- 723
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
- 1,957
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
- 196
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
$$g(x):=\left\{
\matrix{f(x)/x, & x\neq 0 \\
\lim_{x\to 0} f(x)/x, &…
WillG
- 7,382
10
votes
1 answer
Does little Bézout theorem hold for smooth functions?
As a special case of little Bézout theorem, if we have a polynomail $f(x)$ with $f(0)=0$, then there exists another polynomial $g(x)$ such that $f(x)=xg(x)$. It's easy to see that this fact generalizes to analytic functions because we have Taylor…
Syang Chen
- 3,526
10
votes
3 answers
Is the following piecewise-defined function smooth $\in C^\infty(\mathbb{R})$?
Is the function $$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
- 1,957
9
votes
1 answer
Is there a simple way to characterize the smooth functions without using the derivative?
As seen in this question, the derivatives can be easily characterized if we know $C^\infty(\mathbb{R}).$ How can we simply characterize $C^\infty(\mathbb{R})$ if we can't use limits?
mathlander
- 4,097