Transversal intersection. In my textbook, it says: "Consider two curves in the plane, one of which is the x-axis, the other being the graph of a function $f(x)$. The two curves intersect transversally at a point x if $f(x)=0$ (the intersection condition), and $f'(x)\neq0$ (transversality)." I know that: "Two submanifolds of a given finite dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold at that point." From: Transversality (wikipedia). In the example of my textbook, what are the spaces tangent to the x-axis and to the graph of the function $f(x)$? What is the tangent space of the ambient manifold? Thank you very much

The tangent space of the x-axis (at a given point) is again the x-axis (because it's horizontal, i.e. derivative is zero). Then the tangent space of the graph of $f(x)$ is whatever it is, but the key is that when it intersects the x-axis, the tangent space at that intersection point is _not_ horizontal (i.e. not the x-axis). So you have a tangent vector on the x-axis (which is necessarily horizontal) and a tangent vector on the graph of f(x) (which is necessarily not-horizontal), and hence these two vectors are linearly independent, hence span the whole plane (agreeing with Wikipedia)!