could anyone help me with this problem please?
Prove that the definition of continuous function f: R to R is equivalent to the property that for any open set A in R, the inverse-image of A is open.
So I know the definition of a continuous function is that if for every x sub zero in R,and for every epsilon greater than 0, there exists delta such that the absolute value of f(x)-f(x_0) < epsilon if the absolute value of x-x_0 < delta.
I don't know what kind of open set would have the inverse image is also open.
I don't know how to approach this problem
Now say that is a function show that the inverse-image of every open set is open. We want to show it is continous. Let . For consider the set (meaning all that satisfy this inequality). This set, say , is clearly open. So its inverse image is too open. Now trivially. And so by hypothesis there is a such that (meaning all that satisfy this inequality). Thus, . Thus is continous at any point . Q.E.D.