4.10 and 4.14 Theorem in Rudin

Question

1. Could you tell me how part (a) follows from the inequalities (first red box)?
2. Could you tell me why it is not $f(f^{-1}(E)) = E$ (second red box)?

Answer 1

1. Suppose $\mathbf{f}$ is continuous and we want to show that $f_j$ is continuous. Fix $\epsilon>0$, then by continuity of $\mathbf{f}$ there exists $\delta$ such that $|x-y|<\delta$ implies $|\mathbf{f}(x)-\mathbf{f}(y)|<\epsilon$. By chain of inequalities, $$ |f_j(x)-f_j(y)|\leq|\mathbf{f}(x)-\mathbf{f}(y)|<\epsilon. $$ Thus $f_j$ is continuous for each $j=1,\dots,k$. Can you write out the converse?

2. What happens if $f$ is not surjective?