continuity and open subset 
let say  E is a subset of R and f be the real number function o E then
  f is continuou on E if an only if $f^{-1}(V)$ is open in E for every
  open subset $V$ of $R$

what is opn subset V mean? in a continuous function?
I thought open set mean the set contain all its interier points
but what open set mean in he conticuose function
and also what is the closed set mean in continuous function?
 A: Let me rephrase it to (hopefully) make it more clear.

Let's say $E\subseteq\Bbb R$ and $f:E\to\Bbb R.$ The following are equivalent:

*

*The function $f$ is continuous on $E$.

*For every $V\subseteq\Bbb R$ with $V$ open, we have that $f^{-1}(V)$ is (relatively) open in $E$.


When we talk about $V$ being open, we mean exactly that every point of $V$ is an interior point. (Every set contains all its interior points, but open sets don't have any non-interior points.) When we talk about $f^{-1}(V)$ being relatively open in $E,$ we mean that there is a set $U$ that is open (in the usual sense) such that $f^{-1}(V)=U\cap E.$
The notation $f^{-1}(V)$ denotes the preimage or inverse image of the set $V$, and is defined by $$f^{-1}(V):=\{x\in E\mid f(x)\in V\}.$$ Put another way, it is the set of elements of $E$ that $f$ sends to $V$.
A: For a function $f : E \to R$, the notation $f^{-1}(V)$ (for $V \subset R$) is a subset of $E$ defined by
$$
f^{-1}(V) = \{x \in E : f(x) \in V\}.
$$
$f^{-1}(V)$ is a notationally convenient way to describe this set.
