I read the definition of simple arc that an arc which do not cross over itself and is one - one function.
Parabola is not one to one as $ f(x)=x^2 $ implies $f(-1)=f(1)$ but $ -1\ne 1$. Then how parabola is a simple curve?
I am not understanding the meaning of simple arc. Please explain the meaning
1 Answer
$\begingroup$Definition. An arc in the plane is a map $[a,b]\ni t\mapsto (x(t),y(t))\in \Bbb R^2$, (the case $a=-\infty$ and/or $b=+\infty$ is admissible). An arc is said simple if for any $t_1,t_2\in]a,b[$ such that $t_1\neq t_2$ we have$$
(x(t_1),y(t_1))\neq (x(t_2), y(t_2))
$$This definition obviously includes also the standard parabola, since we can parametrize it as follows$$
t\mapsto(x,y)=(x(t),x^2(t))\quad t\in \Bbb R,
$$
where $x=x(t)$ could be any continuous one to one function of $t$, in particular $x=t$.
Now you can see that, according to this definition, the parabola is simple because, when expressed in this form i.e. as a set of ordered couples ("points") in the plane indexed by a real parameter, it is one to one and has no self intersection, since the values of the "independent" variable $x$ which share the same values of the "dependent" variable $y$ are never the same: in your example, having assumed $x=t$, we obviously have$$
(-1,1)\neq (1,1)
$$
