Can an undefined function be said to be not convex?

Question

Is there a formal argument as to whether or not an undefined function can be considered "not convex?"

I think this is practically analogous to saying that I don't have a banana and therefore it doesn't taste good, which in a certain sense makes sense, but only if you're allowed to qualify something that doesn't exist.

An example that came up:

If we have

$f:(0,\infty)\mapsto\mathbb{R}$

$g:(-\infty,0)\mapsto\mathbb{R}$

then can $g \circ f$ be considered "not convex?"

Answer 1

The definition of convexity with which I am familiar is as follows (if you mean some other notion of convexity, please edit your question accordingly):

Let $X$ be an interval in $\mathbb{R}$ (or, more generally, let $X$ be a convex subset of $\mathbb{R}^n$). Then we say that $f : X \to \mathbb{R}$ is convex if $$ f(tx_0 + (1-t)x_1) \le tf(x_0) + (1-t)f(x_1) $$ for all $x_0,x_1 \in X$.

Essentially, this says that the line segment joining any two points in the graph of $f$ must be above the graph of $f$. Note that this definition can be extended to more general domains—I'm not addressing that here.

Now, note that when you say that a function is "undefined," what you are actually saying is that the domain of that function is empty. The emptyset is convex (vacuously), thus we can make sense of the definition of a convex function. So suppose that $f : \emptyset \to \mathbb{R}$ (this function will have an empty range, but the doesn't stop us from making the codomain anything we like). Since there are no $x_0, x_1\in\emptyset$, we make vacuously conclude that $$ f(tx_0 + (1-t)x_1) \le t f(x_0) + (1-t)f(x_1) $$ for all possible $x_0, x_1 \in \emptyset$. Therefore any function with an empty domain will be (vacuously) convex.

For your particular example, if $f : (0,\infty) \to \mathbb{R}$ and $g : \mathbb{R} \to (-\infty, 0)$, then the composition $f\circ g$ is a function with empty domain, i.e. $$ f\circ g : \emptyset \to \mathbb{R}.$$ By the above argument, it is necessarily convex.

Answer 2

Of course, convexity is a topological/geometric idea; even if the function is not regular you can define the concept.

E.G.

$f:\mathbb{R} \to \mathbb{R}$ s.t.

• $f(x)=x^2$ if $x\in \mathbb{Q}$

is not regular but it is Jensen-Convex.