Question

Let \(f:X\rightarrow\mathbb{R} \)be a continuous function on a metric space \(X \)and\(
\quad a, b \in \mathbb{R}\). Prove in a way similar to the proof in the lecture (using the pre image criterion)
c) \(B := \{x \in X : a \leq f(x) \leq b \} \subset X \)is closed

Answer 1

\[
B= f^{-1}([a,b])
\]
and [a,b] is closed.

Answer 2

thank you very much mr Elias