I just learned the concept of Transversality. Briefly, let $f:M \to N$ be a smooth map and $A \subset N$ be a submanifold. If $K \subset M$ we write $f \pitchfork_K A$ to mean that $f$ is transverse to $A$ along $K$, that is, whenever $x\in K$ and $f(x) = y\in A$, the tangent space $T_yN$ is spanned by $T_y A$ and the tangent image of $T_xM$ or $$ f_{*x}(T_xM)+T_yK=T_yN. $$
Obviously, Transversality is a local concept. So I wonder is it possible that $f\pitchfork A$ just on a point but not in any neighborhood of $x$. Formally speaking, Can we choose smooth manifolds $M,N$, a smooth map $f:M \to N$ and a submanifold $A\subset N$ such that there exists a point $x\in M$, $f\pitchfork_x A$ but there is no neighborhood $U$of $x$ satisfying $f\pitchfork_U A$.
Additionnaly, what condition can garuntee such $U$ exists?
Appreciate any help!