Is f(x)=-log(x) a closed function?

Question

I am reading Convex optimization written by Stephen Boyd. In page 640, there is an example said \begin{equation} f(x)=-log(x) \end{equation} is a closed function. But this function seems does not satisfy the definition of closed function in this book. The plot of this function is attached below: And the definition of closed function in the book is attached below:

Can anybody tell me whether this function is closed or not ?

Answer 1

We have $dom\ f = (0,\infty),$ and for any $\alpha \in\mathbb R$ $$ x\in\ dom\ f\ and\ f(x) \leq \alpha \Leftrightarrow x \ge e^{-\alpha} \Leftrightarrow x \in [e^{-\alpha},\infty). $$ Now, the set $[e^{-\alpha},\infty)$ is closed, so $f$ is a closed function, according to the definition.

Answer 2

It's closed. It's level sets are of the form $[b, \infty[$ and hence, closed.