Editing Bump function (section)

==Examples==
[[File:Mollifier Illustration.svg|right|thumb|280px|The 1d bump function <math>\Psi(x).</math>]]

The function <math>\Psi : \mathbb{R} \to \mathbb{R}</math> given by
<math display="block">\Psi(x) = 
\begin{cases}
\exp\left( \frac{1}{x^2 -1}\right), & \text{ if } |x| < 1, \\
0, & \text{ if } |x| \geq 1, 
\end{cases}</math>

is an example of a bump function in one dimension. Note that the support of this function is the closed interval <math> [-1,1]</math>. In fact, by definition of [[Support_(mathematics)|support]], we have that <math> \operatorname{supp}(\Psi):=\overline{\{x\in \mathbb{R}:\Psi(x)\neq 0\}} =\overline{(-1,1)}</math>, where the closure is taken with respect the Euclidean topology of the real line. The proof of smoothness follows along the same lines as for the related function discussed in the [[Non-analytic smooth function]] article. This function can be interpreted as the [[Gaussian function]] <math>\exp\left(-y^2\right)</math> scaled to fit into the unit disc: the substitution <math>y^2 = {1} / {\left(1 - x^2\right)}</math> corresponds to sending <math>x = \pm 1</math> to <math>y = \infty.</math>

A simple example of a (square) bump function in <math>n</math> variables is obtained by taking the product of <math>n</math> copies of the above bump function in one variable, so 
<math display="block">\Phi(x_1, x_2, \dots, x_n) = \Psi(x_1) \Psi(x_2) \cdots \Psi(x_n).</math>

A radially symmetric bump function in <math>n</math> variables can be formed by taking the function <math>\Psi_n : \Reals^n \to \Reals</math> defined by <math>\Psi_n(\mathbf{x})=\Psi(|\mathbf{x}|)</math>. This function is supported on the unit ball centered at the origin.

For another example, take an <math>h</math> that is positive on <math>(c, d)</math> and zero elsewhere, for example 

:<math>h(x) = \begin{cases}
\exp\left(-\frac{1}{(x-c)(d-x)}\right),& c < x < d \\
0,& \mathrm{otherwise}
\end{cases}</math>.


'''Smooth transition functions'''

[[Image:Non-analytic smooth function.png|right|frame|The non-analytic smooth function ''f''(''x'') considered in the article.]]
Consider the function

:<math>f(x)=\begin{cases}e^{-\frac{1}{x}}&\text{if }x>0,\\ 0&\text{if }x\le0,\end{cases}</math>

defined for every [[real number]] ''x''.



[[Image:Smooth transition from 0 to 1.png|right|frame|The smooth transition ''g'' from 0 to 1 defined here.]]
The function

:<math>g(x)=\frac{f(x)}{f(x)+f(1-x)},\qquad x\in\mathbb{R},</math>

has a strictly positive denominator everywhere on the real line, hence ''g'' is also smooth. Furthermore, ''g''(''x'')&nbsp;=&nbsp;0 for ''x''&nbsp;≤&nbsp;0 and ''g''(''x'')&nbsp;=&nbsp;1 for ''x''&nbsp;≥&nbsp;1, hence it provides a smooth transition from the level 0 to the level 1 in the [[unit interval]] <nowiki>[</nowiki>0, 1<nowiki>]</nowiki>. To have the smooth transition in the real interval <nowiki>[</nowiki>''a'', ''b''<nowiki>]</nowiki> with ''a''&nbsp;<&nbsp;''b'', consider the function

:<math>\mathbb{R}\ni x\mapsto g\Bigl(\frac{x-a}{b-a}\Bigr).</math>

For real numbers {{math|''a'' < ''b'' < ''c'' < ''d''}}, the smooth function

:<math>\mathbb{R}\ni x\mapsto g\Bigl(\frac{x-a}{b-a}\Bigr)\,g\Bigl(\frac{d-x}{d-c}\Bigr)</math>

equals 1 on the closed interval <nowiki>[</nowiki>''b'', ''c''<nowiki>]</nowiki> and vanishes outside the open interval (''a'', ''d''), hence it can serve as a bump function.

Caution must be taken since, as example, taking <math>\{a =-1\} < \{b = c =0\}  < \{d=1\}</math>, leads to:
:<math>q(x)=\frac{1}{1+e^{\frac{1-2|x|}{x^2-|x|}}}</math>
which is not an infinitely [[differentiable function]] (so, is not "smooth"), so the constraints {{math|''a'' < ''b'' < ''c'' < ''d''}} must be strictly fulfilled.

Some interesting facts about the function:
:<math>q(x,a)=\frac{1}{1+e^{\frac{a(1-2|x|)}{x^2-|x|}}}</math>
Are that <math>q\left(x,\frac{\sqrt{3}}{2}\right)</math> make smooth transition curves with "almost" constant slope edges (a bump function with true straight slopes is portrayed this [[Delay differential equation|Another example]]).

A proper example of a smooth Bump function would be:
:<math>u(x)=\begin{cases} 1,\text{if } x=0, \\ 0, \text{if } |x|\geq 1, \\ \frac{1}{1+e^{\frac{1-2|x|}{x^2-|x|}}}, \text{otherwise}, \end{cases}</math>

A proper example of a smooth transition function will be:
:<math>w(x)=\begin{cases}\frac{1}{1+e^{\frac{2x-1}{x^2-x}}}&\text{if }0<x<1,\\ 0&\text{if } x\leq 0,\\ 1&\text{if } x\geq 1,\end{cases}</math>

where could be noticed that it can be represented also through [[Hyperbolic functions]]:

:<math>\frac{1}{1+e^{\frac{2x-1}{x^2-x}}} = \frac{1}{2}\left( 1-\tanh\left(\frac{2x-1}{2(x^2-x)} \right) \right)</math>