Fixed points

A function ƒ has a fixed point at x=p if ƒ(p)=p
the function maps the argument to itself.
i.e. the function ƒ intersects the y=x main diagonal line (graph of the identity function f(x)=x) at (p,p)

Infinite number of fixed points:
Exs:
ƒ(x)= x the identity function
tan(x) cot, sec, csc
sin(x) + x

No fixed point. For all x of domain, either f(x)>x or f(x)<x i.e. never crosses or touches main diagonal, is always above it or always below it.
Exs:
linear m=1 (except identity function), ƒ(x)= x+c, c≠0
quadratics: x²+c, c>1/4 -x²-c, c<-1/4
e^x
ln x
cosh x

Fixed point of cosine function is the Dottie number, transcendental, universal attracting fixed point (all numbers upon iteration converge to it).

Quadrisection of the circle from a point on it.
Angles in radians are π/4-D/2 ≈0.4158 and π/4+D/2 ≈1.154
(~23.82° and ~66.17°)