Exercise 6 Let f is continuous on an open interval I prove t

Exercise 6. Let f is continuous on an open interval I, prove that the preimage of I, f^-1(I) = {x, f(x) in I} is open.

Solution

First assume that f is continuous on A, and suppose that c f 1 (V ). Then f(c) V and since V is open it contains an -neighborhood V (f(c)) = (f(c) , f(c) + ) of f(c). Since f is continuous at c, there is a -neighborhood U(c) = (c , c + ) of c such that f (A U(c)) V (f(c)). This statement just says that if |x c| < and x A, then |f(x) f(c)| < . It follows that A U(c) f 1 (V ), meaning that f 1 (V ) contains a relative neighborhood of c. Therefore f 1 (V ) is relatively open in A. Conversely, assume that f 1 (V ) is open in A for every open V in R, and let c A. Then the preimage of the -neighborhood (f(c), f(c) +) is open in A, so it contains a relative -neighborhood A(c, c+). It follows that |f(x)f(c)| < if |x c| < and x A, which means that f is continuous at c. As one illustration of how we can use this result, we prove that continuous functions map intervals to intervals. this is a special case of the fact that continuous functions map connected sets to connected sets (s