Flat function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially <a href="/facts/Real_analysis/12WjlXCW">real analysis</a>, a <a href="/facts/Real_function/bbNG3DnN">real function</a> is flat at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 if all its <a href="/facts/Higher-order_derivative/7Ito70Dh">derivatives</a> at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 exist and equal 0. 
A function that is flat at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 is not <a href="/facts/Analytic_function/JhL3z8FP">analytic</a> at 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 unless it is <a href="/facts/Constant_function/BmPx6dn7">constant</a> in a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 (since an analytic function must equals the sum of its <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a>).
An example of a flat function at 0 is the function such that 
 
 
 
 f
 (
 0
 )
 =
 0
 
 
 {\displaystyle f(0)=0}
 
 and 
 
 
 
 f
 (
 x
 )
 =
 
 e
 
 −
 1
 
 /
 
 
 x
 
 2
 
 
 
 
 
 
 {\textstyle f(x)=e^{-1/x^{2}}}
 
 for 
 
 
 
 x
 ≠
 0.
 
 
 {\displaystyle x\neq 0.}

The function need not be flat at just one point. Trivially, <a href="/facts/Constant_function/BmPx6dn7">constant functions</a> on 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 are flat everywhere. But there are also other, less trivial, examples; for example, the function such that 
 
 
 
 f
 (
 x
 )
 =
 0
 
 
 {\displaystyle f(x)=0}
 
 for 
 
 
 
 x
 ≤
 0
 
 
 {\displaystyle x\leq 0}
 
 and 
 
 
 
 f
 (
 x
 )
 =
 
 e
 
 −
 1
 
 /
 
 
 x
 
 2
 
 
 
 
 
 
 {\textstyle f(x)=e^{-1/x^{2}}}
 
 for 
 
 
 
 x
 >
 0.
 
 
 {\displaystyle x>0.}

Flat function open-in-new

Flat function