Inmathematics, afunction is said tovanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined onnormed vector spaces and the other applying to functions defined onlocally compact spaces. Aside from this difference, both of these notions correspond to the intuitive notion of adding a point at infinity, and requiring the values of the function to get arbitrarily close to zero as one approaches it. This definition can be formalized in many cases by adding an (actual)point at infinity.

A function on anormed vector space is said tovanish at infinity if the function approaches${\displaystyle 0}$ as the input grows without bounds (that is,${\displaystyle f(x)\to 0}$ as${\displaystyle \Vert x\Vert \to \mathrm{\infty }}$). Or,

${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}f(x)=\underset{x\to +\mathrm{\infty }}{lim}f(x)=0}$

in the specific case of functions on the real line.

For example, the function

${\displaystyle f(x)=\frac{1}{x^2+1}}$

defined on thereal line vanishes at infinity.

Alternatively, a function${\displaystyle f}$ on alocally compact space${\displaystyle \mathrm{\Omega }}$vanishes at infinity, if given anypositive number${\displaystyle \epsilon >0}$, there exists acompact subset${\displaystyle K\subseteq \mathrm{\Omega }}$ such that

whenever the point${\displaystyle x}$ lies outside of${\displaystyle K.}$^[1][2] In other words, for each positive number${\displaystyle \epsilon >0}$, the set${\displaystyle \left\{x\in \mathrm{\Omega }:\Vert f(x)\Vert \ge \epsilon \right\}}$ has compact closure. For a given locally compact space${\displaystyle \mathrm{\Omega }}$ theset of such functions

${\displaystyle f:\mathrm{\Omega }\to \mathbb{K}}$

valued in${\displaystyle \mathbb{K},}$ which is either${\displaystyle \mathbb{R}}$ or${\displaystyle \mathbb{C},}$ forms a${\displaystyle \mathbb{K}}$-vector space with respect topointwisescalar multiplication andaddition, which is often denoted${\displaystyle C_0(\mathrm{\Omega }).}$

As an example, the function

${\displaystyle h(x,y)=\frac{1}{x+y}}$

where${\displaystyle x}$ and${\displaystyle y}$ arereals greater or equal 1 and correspond to the point${\displaystyle (x,y)}$ on${\displaystyle {\mathbb{R}}_{\ge 1}^2}$ vanishes at infinity.

Anormed space is locally compact if and only if it is finite-dimensional so in this particular case, there are two different definitions of a function "vanishing at infinity". The two definitions could be inconsistent with each other: if${\displaystyle f(x)=\Vert x{\Vert }^{-1}}$ in an infinite dimensionalBanach space, then${\displaystyle f}$ vanishes at infinity by the${\displaystyle \Vert f(x)\Vert \to 0}$ definition, but not by the compact set definition.

Refining the concept, one can look more closely to therate of vanishing of functions at infinity. One of the basic intuitions ofmathematical analysis is that theFourier transform interchangessmoothness conditions with rate conditions on vanishing at infinity. UsingbigO notation, therapidly decreasing test functions oftempered distribution theory aresmooth functions that are

${\displaystyle O\left(|x{|}^{-N}\right)}$

for all${\displaystyle N}$, as${\displaystyle |x|\to \mathrm{\infty }}$, and such that all theirpartial derivatives satisfy the same condition too. This condition is set up so as to be self-dual under Fourier transform, so that the correspondingdistribution theory oftempered distributions will have the same property.

• Infinity – Mathematical concept
• Projectively extended real line – Real numbers with an added point at infinity
• Zero of a function – Point where function's value is zero

• Hewitt, E and Stromberg, K (1963).Real and abstract analysis. Springer-Verlag.{{cite book}}: CS1 maint: multiple names: authors list (link)

• Mathematical analysis

• CS1 maint: multiple names: authors list