Flat function
= Flat function
{title2=$f^{(n)}(a)=0\ (n\geq0)$}
A <smooth function> is flat at $a$ when every <derivative>, including order zero, vanishes at $a$. The function $e^{-1/x^2}$ off zero, extended by zero at zero, is flat there but not identically zero near zero. Thus its <Taylor series> does not determine it. A <real analytic function> that is flat at a point vanishes on a neighborhood of that point.