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)
a) $A:=\{x\in X : f(x) \leq b\} \subset X$ 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)
a) $A:=\{x\in X : f(x) \leq b\} \subset X$ is closed

anonymous · Answer

The interval
$$
A= f^{-1} (]-\infty, b])
$$

anonymous · Answer

$$]-\infty, b]
$$
is closed.

anonymous · Answer

yes...

anonymous · Answer

Mr Elias do you think is it enough for this question ? thx

anonymous · Answer

Yes,
The inverse image of a closed set by a continuous function is closed.

anonymous · Answer

thank you very much, i have one queston more from last week but i must find it

anonymous · Answer

i am sorry Mr Elias :) i forget it

anonymous · Answer

Close this question and ask a new one.