Exercises
[2BB](Proposed on 2022-12) Consider this statement.
«Let \(f:X\to Y\) and \(x_ 0\in X\), then \(f\) is continuous at \(x_ 0\) when, for every open set \(B\subseteq Y\) with \(f(x_ 0)\in B\), we have that \(f^{-1}(B)\) is open.»
This statement is incorrect.
Build an example of a function \(f:{\mathbb {R}}\to {\mathbb {R}}\) that is continuous at \(x_ 0=0\) but such that, for every \(J=(a,b)\) open non-empty bounded interval, \(f^{-1}(J)\) is not open.
Solution 1