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r displaystyle x y in mathbb r 2 mapsto psi r where r x 2 y 2 1 2 displaystyle r left x 2 y 2 right 1 2 and ψ r e 1 1 r 2 1 r 1 displaystyle psi r e 1 1 r 2 cdot mathbf 1 _ r 1 in mathematical analysis a bump function is a localized auxiliary function usually chosen to be smooth and to have compact support bump functions are commonly used as cutoff functions for example functions that are equal to 1 on a prescribed set and vanish outside a larger set and as standard examples of kernels used to construct mollifiers some authors use the term more broadly for any compactly supported smooth function such functions are important examples of test functions especially in distribution theory but the terms bump function and test function are not synonymous in all contexts examples edit the 1d bump function ψ x displaystyle psi x the function ψ r r displaystyle psi mathbb r to mathbb r given by ψ x exp 1 x 2 1 if x 1 0 if x 1 displaystyle psi x begin cases exp left frac 1 x 2 1 right text if x 1 0 text if x geq 1 end cases is an example of a bump function in one dimension note that the support of this function is the closed interval 1 1 displaystyle 1 1 in fact by definition of support we have that supp ψ x r ψ x 0 1 1 displaystyle operatorname supp psi overline x in mathbb r psi x neq 0 overline 1 1 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 exp y 2 displaystyle exp left y 2 right scaled to fit into the unit disc the substitution y 2 1 1 x 2 displaystyle y 2 1 left 1 x 2 right corresponds to sending x 1 displaystyle x pm 1 to y displaystyle y infty a simple example of a square bump function in n displaystyle n variables is obtained by taking the product of n displaystyle n copies of the above bump function in one variable so φ x 1 x 2 x n ψ x 1 ψ x 2 ψ x n displaystyle phi x_ 1 x_ 2 dots x_ n psi x_ 1 psi x_ 2 cdots psi x_ n a radially symmetric bump function in n displaystyle n variables can be formed by taking the function ψ n r n r displaystyle psi _ n mathbb r n to mathbb r defined by ψ n x ψ x displaystyle psi _ n mathbf x psi mathbf x this function is supported on the unit ball centered at the origin for another example take an h displaystyle h that is positive on c d displaystyle c d and zero elsewhere for example h x exp 1 x c d x c x d 0 o t h e r w i s e displaystyle h x begin cases exp left frac 1 x c d x right c x d 0 mathrm otherwise end cases smooth transition functions the non analytic smooth function f x considered in the article a standard starting point is the function f x e 1 x if x 0 0 if x 0 displaystyle f x begin cases e frac 1 x text if x 0 0 text if x leq 0 end cases defined for every real number x the smooth transition g from 0 to 1 defined here from this define g x f x f x f 1 x x r displaystyle g x frac f x f x f 1 x qquad x in mathbb r the denominator is strictly positive everywhere on the real line so g is smooth moreover g x 0 for x 0 and g x 1 for x 1 so g gives a smooth transition from 0 to 1 on the unit interval 0 1 rescaling gives a smooth transition on any interval a b with a b r x g x a b a displaystyle mathbb r ni x mapsto g bigl frac x a b a bigr a compactly supported bump function can be obtained by multiplying a rising transition by a falling transition for real numbers a b c d the function r x g x a b a g d x d c displaystyle mathbb r ni x mapsto g bigl frac x a b a bigr g bigl frac d x d c bigr is smooth equals 1 on the closed interval b c and vanishes outside the open interval a d thus it can serve as a bump function when b c the plateau has positive length when b c it degenerates to a single point but the function is still smooth for example taking a 1 displaystyle a 1 b c 0 displaystyle b c 0 and d 1 displaystyle d 1 gives the smooth bump function u x 1 if x 0 0 if x 1 1 1 e 1 2 x x 2 x otherwise displaystyle 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 the formula q x 1 1 e 1 2 x x 2 x displaystyle q x frac 1 1 e frac 1 2 x x 2 x is the expression for this function only on 0 x 1 displaystyle 0 x 1 by itself it does not specify the endpoint values at x 1 0 1 displaystyle x 1 0 1 a related parameterized interior expression is q x a 1 1 e a 1 2 x x 2 x displaystyle q x a frac 1 1 e frac a 1 2 x x 2 x for example q x 3 2 displaystyle q left x frac sqrt 3 2 right with suitable endpoint values supplied gives smooth transition curves with almost constant slope edges a bump function with true straight slopes is portrayed by this example the transition function g above can also be written explicitly as w x 1 1 e 2 x 1 x 2 x if 0 x 1 0 if x 0 1 if x 1 displaystyle 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 and on 0 x 1 displaystyle 0 x 1 its nonconstant branch can be represented using hyperbolic functions 1 1 e 2 x 1 x 2 x 1 2 1 tanh 2 x 1 2 x 2 x displaystyle 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 notice that u x displaystyle u x could be used to built extended flat top smooth bump functions as example for an integer value m 0 displaystyle m 0 y x k m m u x k displaystyle y x sum _ k m m u x k will have an unitary flat top on x m m displaystyle x in m m while being zero on x m 1 displaystyle x geq m 1 somehow working similar to a partition of unity existence of bump functions edit an illustration of the sets in the construction it is possible to construct bump functions to specifications stated formally if k r n displaystyle k subset mathbb r n is compact and u r n displaystyle u subset mathbb r n is an open set containing k displaystyle k there exists a bump function ϕ displaystyle phi which is 1 displaystyle 1 on k displaystyle k and 0 displaystyle 0 outside of u displaystyle u since u displaystyle u can be taken to be a very small neighborhood of k displaystyle k this amounts to being able to construct a function that is 1 displaystyle 1 on k displaystyle k and falls off rapidly to 0 displaystyle 0 outside of k displaystyle k while still being smooth bump functions defined in terms of convolution the construction proceeds as follows one considers a compact neighborhood v displaystyle v of k displaystyle k contained in u displaystyle u so k v v u displaystyle k subseteq v circ subseteq v subseteq u the characteristic function χ v displaystyle chi _ v of v displaystyle v will be equal to 1 displaystyle 1 on v displaystyle v and 0 displaystyle 0 outside of v displaystyle v so in particular it will be 1 displaystyle 1 on k displaystyle k and 0 displaystyle 0 outside of u displaystyle u this function is not smooth however the key idea is to smooth χ v displaystyle chi _ v a bit by taking the convolution of χ v displaystyle chi _ v with a mollifier the latter is just a bump function with a very small support and whose integral is 1 displaystyle 1 such a mollifier can be obtained for example by taking the bump function φ displaystyle phi from the previous section and performing appropriate scalings bump functions defined in terms of a function c r 0 displaystyle c mathbb r to 0 infty with support 0 displaystyle infty 0 an alternative construction that does not involve convolution is now detailed it begins by constructing a smooth function f r n r displaystyle f mathbb r n to mathbb r that is positive on a given open subset u r n displaystyle u subseteq mathbb r n and vanishes off of u displaystyle u 1 this function s support is equal to the closure u displaystyle overline u of u displaystyle u in r n displaystyle mathbb r n so if u displaystyle overline u is compact then f displaystyle f is a bump function start with any smooth function c r r displaystyle c mathbb r to mathbb r that vanishes on the negative reals and is positive on the positive reals that is c 0 displaystyle c 0 on 0 displaystyle infty 0 and c 0 displaystyle c 0 on 0 displaystyle 0 infty where continuity from the left necessitates c 0 0 displaystyle c 0 0 an example of such a function is c x e 1 x displaystyle c x e 1 x for x 0 displaystyle x 0 and c x 0 displaystyle c x 0 otherwise 1 fix an open subset u displaystyle u of r n displaystyle mathbb r n and denote the usual euclidean norm by displaystyle cdot so r n displaystyle mathbb r n is endowed with the usual euclidean metric the following construction defines a smooth function f r n r displaystyle f mathbb r n to mathbb r that is positive on u displaystyle u and vanishes outside of u displaystyle u 1 so in particular if u displaystyle u is relatively compact then this function f displaystyle f will be a bump function if u r n displaystyle u mathbb r n then let f 1 displaystyle f 1 while if u displaystyle u varnothing then let f 0 displaystyle f 0 so assume u displaystyle u is neither of these let u k k 1 displaystyle left u_ k right _ k 1 infty be an open cover of u displaystyle u by open balls where the open ball u k displaystyle u_ k has radius r k 0 displaystyle r_ k 0 and center a k u displaystyle a_ k in u then the map f k r n r displaystyle f_ k mathbb r n to mathbb r defined by f k x c r k 2 x a k 2 displaystyle f_ k x c left r_ k 2 left x a_ k right 2 right is a smooth function that is positive on u k displaystyle u_ k and vanishes off of u k displaystyle u_ k 1 for every k n displaystyle k in mathbb n let m k sup p f k p 1 x 1 p n x n x x r n and p 1 p n z satisfy 0 p i k and p i p i displaystyle m_ k sup left left frac partial p f_ k partial p_ 1 x_ 1 cdots partial p_ n x_ n x right x in mathbb r n text and p_ 1 ldots p_ n in mathbb z text satisfy 0 leq p_ i leq k text and p sum _ i p_ i right where this supremum is not equal to displaystyle infty so m k displaystyle m_ k is a non negative real number because r n u k u k r n displaystyle left mathbb r n setminus u_ k right cup overline u_ k mathbb r n the partial derivatives all vanish equal 0 displaystyle 0 at any x displaystyle x outside of u k displaystyle u_ k while on the compact set u k displaystyle overline u_ k the values of each of the finitely many partial derivatives are uniformly bounded above by some non negative real number note 1 the series f k 1 f k 2 k m k displaystyle f sum _ k 1 infty frac f_ k 2 k m_ k converges uniformly on r n displaystyle mathbb r n to a smooth function f r n r displaystyle f mathbb r n to mathbb r that is positive on u displaystyle u and vanishes off of u displaystyle u 1 moreover for any non negative integers p 1 p n z displaystyle p_ 1 ldots p_ n in mathbb z 1 p 1 p n p 1 x 1 p n x n f k 1 1 2 k m k p 1 p n f k p 1 x 1 p n x n displaystyle frac partial p_ 1 cdots p_ n partial p_ 1 x_ 1 cdots partial p_ n x_ n f sum _ k 1 infty frac 1 2 k m_ k frac partial p_ 1 cdots p_ n f_ k partial p_ 1 x_ 1 cdots partial p_ n x_ n where this series also converges uniformly on r n displaystyle mathbb r n because whenever k p 1 p n displaystyle k geq p_ 1 cdots p_ n then the k displaystyle k th term s absolute value is m k 2 k m k 1 2 k displaystyle leq tfrac m_ k 2 k m_ k tfrac 1 2 k this completes the construction as a corollary given two disjoint closed subsets a b displaystyle a b of r n displaystyle mathbb r n the above construction guarantees the existence of smooth non negative functions f a f b r n 0 displaystyle f_ a f_ b mathbb r n to 0 infty such that for any x r n displaystyle x in mathbb r n f a x 0 displaystyle f_ a x 0 if and only if x a displaystyle x in a and similarly f b x 0 displaystyle f_ b x 0 if and only if x b displaystyle x in b then the function h f a f a f b r n 0 1 displaystyle h frac f_ a f_ a f_ b mathbb r n to 0 1 is smooth and for any x r n displaystyle x in mathbb r n h x 0 displaystyle h x 0 if and only if x a displaystyle x in a h x 1 displaystyle h x 1 if and only if x b displaystyle x in b and 0 h x 1 displaystyle 0 h x 1 if and only if x a b displaystyle x not in a cup b 1 in particular h x 0 displaystyle h x neq 0 if and only if x r n a displaystyle x in mathbb r n smallsetminus a so if in addition u r n a displaystyle u mathbb r n smallsetminus a is relatively compact in r n displaystyle mathbb r n where a b displaystyle a cap b varnothing implies b u displaystyle b subseteq u then h displaystyle h will be a smooth bump function with support in u displaystyle overline u properties and uses edit while bump functions are smooth the identity theorem prohibits them from being analytic unless they vanish identically bump functions are often used as mollifiers as smooth cutoff functions and to form smooth partitions of unity they are the most common class of test functions used in analysis the space of bump functions is closed under many operations for instance the sum product or convolution of two bump functions is again a bump function and any differential operator with smooth coefficients when applied to a bump function will produce another bump function if the boundaries of the bump function domain is x displaystyle partial x to fulfill the requirement of smoothness it has to preserve the continuity of all its derivatives which leads to the following requirement at the boundaries of its domain lim x x d n d x n f x 0 for all n 0 n z displaystyle lim _ x to partial x pm frac d n dx n f x 0 text for all n geq 0 n in mathbb z the fourier transform of a bump function is a real analytic function and it can be extended to the whole complex plane hence it cannot be compactly supported unless it is zero since the only entire analytic bump function is the zero function see paley wiener theorem and liouville s theorem because the bump function is infinitely differentiable its fourier transform must decay faster than any finite power of 1 k displaystyle 1 k for a large angular frequency k displaystyle k 2 the fourier transform of the particular bump function ψ x e 1 1 x 2 1 x 1 displaystyle psi x e 1 1 x 2 mathbf 1 _ x 1 from above can be analyzed by a saddle point method and decays asymptotically as k 3 4 e k displaystyle k 3 4 e sqrt k for large k displaystyle k 3 the integral of the bump function ψ x displaystyle psi x is given by ψ x d x e 1 2 k 1 1 2 k 0 1 2 displaystyle int _ infty infty psi x dx e 1 2 left k_ 1 left frac 1 2 right k_ 0 left frac 1 2 right right where k 1 x displaystyle k_ 1 x and k 0 x displaystyle k_ 0 x are the modified bessel functions of the second kind 4 see also edit cutoff function integration kernels for smoothing out sharp features pages displaying short descriptions of redirect targets laplacian of the indicator limit of sequence of smooth functions non analytic smooth function mathematical functions which are smooth but not analytic schwartz space function space of all functions whose derivatives are rapidly decreasing flat function function whose all derivatives vanish at a point citations edit the partial derivatives p f k p 1 x 1 p n ...