functions 2

In preceding post, we defined a function as follows.

A relation $f$  from a member of a set $X$  to a set $Y$  which is many to one or one to one is called a function from $X$  to $Y$. Then, we write
$f:X\rightarrow Y$

There are usually two metric spaces $S,T$ , and $X\subset S$  and $Y\subset T$ are assumed.
$Y=f(X)=\left\{ f(x) | x\in X \right\}$  is the range or the image of the function.
Conversely, the inverse image of the function is
$f^{-1}(Y)=\left\{ x | f(x)\in Y \right\}$
As it is the domain or a part of the domain of the function, $f^{-1}(Y)\subset X$

By using the distance function $d$  on $S$  and $d'$  on$T$ , we shall define again continuous functions by epsilon-delta proofs.

A function $f$  is continuous at a point c if, for any $\epsilon>0$, there is a $\delta>0$
such that if $d(x,c)<\delta$ , $d'(f(x),f(c))<\epsilon$ 

Next definition is derived from above. But no distance fuctions are used.

A function is continuous if and only if the inverse image of a open set is open.

Similarly, the definition by a closed set is available.

A function is continuous if and only if the inverse image of a closed set is closed.

Two definitions are equivalent each other. However these are not true on the range or the image.