Is there a function that is continuously differentiable but not proper?

Question

A convex function $f$ is called proper if \begin{equation} \begin{aligned} \exists x ,f(x)<+\infty,\\ \forall x ,f(x)>-\infty. \end{aligned} \end{equation}

My question is:

Is it possible that $f$ is continuously differentiable but not proper?

Answer 1

It depends on your definition of "continuously differentiable".

In a certain sense, the function $f \colon \mathbb R \to \bar{\mathbb R}$ with $f(x) = -\log(x)$ for $x > 0$, $f(x) = \infty$ for $x \le 0$ qualifies. In fact:

• We could say that it is continuous, because $\lim_{y \to x} f(y) = f(x)$ holds for all $x \in \mathbb R$, if we allow $+\infty$ as a limit.
• It is differentiable at all $x \in \operatorname{dom} f$. This is the best we can hope for, since the definition of differentiability at $x$ needs $f(x) < \infty$.
• The derivative $f' \colon \operatorname{dom} f \to \mathbb R$ is continuous.