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

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

anonymous · Answer

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

anonymous · Answer

thank you very much mr Elias