TITLE: Reflection of light from function graph
QUESTION [20 upvotes]: Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$  and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ y_0>0,\,y_0 <f(x_0).$  Light is assumed to be reflected from the plot $y=f(x)$ and the $x$-axis. Does there exist a number $R$ s.t. the part of the graph $y=f(x)$
 for $x>R$ is not lightened? 
The model example $f(x):=e^{-x},\,P(0,0.5)$ suggests the answer is yes.
The question is migrated from SE.

REPLY [3 votes]: In the paper The existence of unbounded oscillating trajectories in a problem of billiards (1962) Leontovich proved that under bell-like curve (it must be zero at  $\pm\infty$) each trajectory oscillates, i.e. it crosses y-axis infinitely often. Also he proved that among all trajectories do exist finite and infinite ones.