Flat function along a set

A smooth function $f:U\subseteq\mathbb R^n\to\mathbb R^m$ is flat on $A\subseteq U$ if

\partial^\alpha f(a)=0\qquad(a\in A,\ \alpha\in\mathbb N_0^n).

Thus every finite Cartesian jet vanishes on $A$ . The derivative of order zero is included, so $f$ itself vanishes there.

Boundary convention

For a function smooth up to an endpoint from one side, flatness at that endpoint means that every one-sided derivative extends continuously with value zero. With spatial parameters, all mixed derivatives are included and their limits are locally uniform in those parameters.

Flat need not mean locally zero

The flat exponential is positive on one side of zero yet flat at zero. In contrast, a real-analytic function flat at a point is zero in a neighborhood of that point, because its convergent Taylor series is zero.