Understanding Inverse Images and Continuity in Real Analysis

[im2fastfouru]

1. a) Show that (f^-1 S)compliment = f^-1(S compliment) for any set S of reals.


Then use part a) to show The function f is continuous iff f^-1(S) is closed for every closed set S.



2. inverse image = f^-1(S) = {x: f(x) \in S}
f is continuous iff for every open set U \in the reals, f^-1(U) is open.

 

[ircdan]

part a is true for any set, so you need to show f^-1(S^c) = (f^-1(S))^c,

so take x in f^-1(S^c), so f(x) is in S^c, so f(x) is not in S, so ...

for part b, what have you tried and what is your definition of continuous, delta/epsilon?