There exists a differentiable function $f\colon\mathbb{R}\to\mathbb{R}$ with the following property:
the tangent at each point has infinitely many common points with the graph
/Edit: $f$ nonlinear/
For $\sin x^2$ we have this property at point $0$, but it's hard to imagine this could happen at every point… I think the statement is false, but no idea how to prove it.
Best Answer
$e^x\sin(x)$ meets any straight line infinitely many times.
More generally, $f(x)g(x)$ where $f$ has a superlinear growth and $g$ is periodic and alternates.