Finding the points of intersection

Question

Let f(x) be a real valued function defined for all real numbers s such that $|f(x)-f(y)| \leq (1/2)|x-y|$ for all $x,y$. Then what is the number of points of intersection of the graph of $y = f(x)$ and the line $y=x$?

Answer 1

$f(x)$ is a contraction, so it has a unique fixed point by the contraction mapping theorem, that is a point where $f(x)=x$. So there is exactly one point where $y=f(x)=x$.