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9 votes
Accepted

Formula for bump function

I have a stricter version than what is being asked for. I have had a very similar question in the past nag at me but never figured it out. What I had been interested in was more restrictive than OP ...
Cameron Williams's user avatar
8 votes

Formula for bump function

If you only require the first order derivative to be continuous, then you can take $$ f(x) = \begin{cases} \cos^2\frac{\pi x}{2}&\text{when $-1\leq x\leq 1$}\\ 0,&\text{otherwise} \end{cases} $...
md2perpe's user avatar
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8 votes

Formula for bump function

When you raise your function to some power $p > 0$, you can "lift" the graph up or down and change the integral this way while still having $g(0) = 1$, i.e. consider instead $$ g(x)^p = \...
ljfa's user avatar
  • 503
6 votes

Formula for bump function

I have used the $C_C^\infty$ function $$ f(x)=\left\{\begin{array}{}e^{-\frac4\pi\tan^2(\pi x/2)}\sec^2(\pi x/2)&\text{for $|x|\lt1$}\\0&\text{for $|x|\ge1$}\end{array}\right. $$ $$ \begin{...
robjohn's user avatar
  • 348k
5 votes

Limit of a function when 'a' is not in the domain

In a limit $\lim_{x\to a} f(x)$, the number $a$ need not be in the domain of the function, but it does need to be a "limit point" of the domain, i.e. you need to be able to get arbitrarily ...
J. Chapman's user avatar
  • 1,102
4 votes
Accepted

Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $

Since $\pi(N)\sim N/\ln N$ by the prime number theorem, $$ f(x) = \lim_{N\rightarrow\infty}\frac{\pi(Nx)}{\pi(N)} = \lim_{N\rightarrow\infty}\frac{Nx/\ln(Nx)}{N/\ln N} = \lim_{N\rightarrow\infty}\frac{...
Einar Rødland's user avatar
4 votes
Accepted

Solutions to $(f(x)-f(y))^3=f\left(x^3\right)-f\left(y^3\right)$

I leave my proof, let me know if there are some incorrect reasonings please. The globally constant functions satisfy our equation, so we can suppose there are $x$ and $y$ such that $f(x)\neq f(y)$. We ...
Federico Fallucca's user avatar
3 votes

Formula for bump function

Let $f(x)=\frac{\left(x^2+1\right)\text{sech}^2\left(\frac{2x}{1-x^2}\right)}{\left(1-x^2\right)^2}$ on $[-1,1]$. This was found by constructing the anti-derivative $F(x)=\frac{1}{2}\text{tanh}\left(\...
Andrew Fillmore's user avatar
2 votes

Formula for bump function

You can do this simply by stitching together four cubic curves. Start with the segment of the curve $f(x)=1-4x^3$ on the interval $[0,\frac12]$: Rotate this curve about the point $(\frac12,\frac12)$, ...
TonyK's user avatar
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1 vote

Is there an analog for factorials in division, and if so, what are its applications and properties?

As some of us have indicated in the comments to the OP, repeated division without grouping is ambiguous because division is not associative. That is, $$ (a \div b) \div c \not= a \div (b \div c) $$ ...
Brian Tung's user avatar
  • 34.5k
1 vote

Formula for bump function

Also consider this. I expect the code is clear enough to adapt another platforms. Attending the comment, all we can see
janmarqz's user avatar
  • 10.6k
1 vote

Formula for bump function

A pretty simple one is $f(x) := (1 - x^2)^\alpha$ on $[-1, 1]$ and zero elsewhere, where $\alpha := 2.38175026...$, chosen to enforce the total integral constraint.
A rural reader's user avatar
1 vote

A question from Advanced problems in mathematics for JEE main and advanced aka black book from the topic of functions

By AM-GM, LHS $\ge 2 \sqrt{16^{x^2+y^2+x+y}}$ Now $x^2+y^2+x+y=\left(x+\dfrac{1}{2}\right)^2+\left(y+\dfrac{1}{2}\right)^2 -\dfrac{1}{2} \ge -\dfrac{1}{2}$ Hence $2 \sqrt{16^{x^2+y^2+x+y}} \ge 1$ with ...
Hari Shankar's user avatar
  • 3,706
1 vote

Continuity of an oscillatory type function

It is undefined. The question that we usually like to ask, as adapted to this situation, is the following: Is there some real number $c$ such that if we define $$f(x)=\begin{cases} c & x=0 \\ x \...
Ian's user avatar
  • 102k

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