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)

b) $U := \{x \in X : f(x) a \} \subset X$ is open

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)

b) $U := \{x \in X : f(x) > a \} \subset X$ is open

anonymous · Answer

$$
U= f^{-1}(]a, \infty[)
$$
and
$$
]a, \infty[
$$
is open.

anonymous · Answer

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anonymous · Answer

yw