Fixed points: A self explanatory discussion on fixed points. 



Fixed point also known as invariant point is a beautiful concept of mathematics. The fixed point of a function is defined as an element of the function's domain that is mapped to itself by the function domain, very beautiful although with a rather complicate grammar.
This can be written as, a function $f$ defined on the real line is said to have a fixed point at $a$ if $f(a)=a$. A collection of fixed points is called a fixed set.
It is important to note the difference between a fixed point and a stationary point. In stationary points, if $a\in\mathbb{R}$ and $f$ is a function defined on $\mathbb{R}$, then $f'(a)=0$.
An example of functions that have fixed points are:
  1. If $f(x)=x^2-6$, then $3$ is a fixed point because $f(3)=3$.
  2.  Also take the function $f(x)=x^2-3x+4$, $2$ is fixed point as $f(2)=2$.
There are many other functions with fixed points, although not all functions have fixed points. There are quite a large number of functions that don't have a fixed point in $\mathbb{R}$. 
Points that come back to the same value after a finite number of iterations of the function are called periodic points. 
A fixed point is a periodic point with period equal to one. 
There is a type of fixed point called the attracting fixed point, which is defined as:
An attracting fixed point of a function $f$ has a fixed point at $x_0$ such that for any value of $x$ in the domain that is close enough to $x_0$, the iterated function $x,f(x),f(f(x)),f(f(f(x))),...$ converges to $x_0$.
Also, we must note that not all fixed point are attracting. 
The natural cosine function is an example of an attracting fixed point. 
One area in mathematics where the study of fixed point has been very progressive is in topology. The concept of fixed points in topology is known as topological fixed points. 
In topological fixed point, a topological space $X$ is said to have the fixed point property (FPP) if for any continuous function $f(x)=x$. Homomorphisms in topological spaces preserve the fixed point property since they are continuous. A fixed point is a periodic point with period equal to one