Suppose that f : X → ]−∞, +∞] is convex and proper, that x ̄ ∈ dom f

Question

Suppose that $f : X → ]−∞, +∞]$ is convex and proper, that $x ̄ ∈ dom f$ , and that $λ > 0.$ Show that

$$∂(λf)(x ̄) = λ∂f(x ̄).$$

I am looking for a little help with this one. I am a bit unclear as to what is meant by 'proper'. Also I have the property $$v∈∂f(x ̄) ⇔ (∀x∈X)f(x ̄)+⟨v,x−x ̄⟩≤f(x),$$ but I'm not sure how this helps me.

Any hints or suggestions is greatly appreciated.

Answer 1

Let me briefly sketch the proof. Take $v \in \lambda \, \partial f(\bar x)$. Write down the definition of $v/\lambda \in \partial f(\bar x)$. Manipulate the expression to find $v \in \partial(\lambda f)(\bar x)$. Conclude.