TITLE: Is f(x)=-log(x) a closed function?
QUESTION [5 upvotes]: 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 ?

REPLY [6 votes]: 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.

REPLY [3 votes]: It's closed. It's level sets are of the form $[b, \infty[$ and hence, closed.