Consider the $C^1$ function $f: [0,1] \to \mathbb{R}$. I understand that a curve in the plane that intersects the graph of $f$ non-transversally would be tangent to it at a point of intersection. I also understand that transversal intersection of submanifolds is generic. My question is, by viewing the graphs of two functions as submanifolds in $\mathbb{R}^2$ can I then state the following:
The set of all functions in $g \in C^1([0,1])$ whose graph intersects the graph of $f$ non-transversally are meagre?
I suspect this is a very simple question; I'm relatively new to differential topology and was hoping somebody could set me straight with some intuition!