Maximum of nothing

Question

Say I have some function where a certain condition cannot be fulfilled for certain inputs, e.g. $$f(x)=\max_{a\in \mathbb{N}, a < x}\quad a^2$$ and I plug in $x=-3$. Then there is no number $a$ such that $a\in \mathbb{N}$ and $a<-3$, hence I cannot compute $a^2$. What is $f(-3)$?

Answer 1

The example you provide is undefined at the point $x=-3$. Therefore, if you were to plot this as a graph there would be no defined value of $f$ corresponding to $x$ at this point.

There are many such examples of functions that are undefined at specific values of $x$. For example $h(x) = \frac{1}{x}$ is undefined at $x=0$ or $g(x) = \log (x)$ is undefined for all negative values of $x$.

There are a number of ways you can deal with this. Either you can restrict the domain so that $x$ can only take values where the maximum is defined. Or, you can keep an unrestricted domain and allow there to be discontinuities in your function. Another common fix is to multiply the function by an indicator function so that the function is equal to $0$ at all the points which were previously undefined.

Also, as best practice for notation, you should define the function in the question in the following way. For $f: \mathbb{R} \rightarrow \mathbb{R}$ let the function $$f(x) := \max \{ a^2 \in \mathbb{N} \space : \space a<x, \space a \in \mathbb{N}\}$$

You should exclude negative value of $x$ to ensure that the function excludes the discontinuity that you provided by simply rewriting the function as $f: \mathbb{R}^+ \rightarrow \mathbb{R}$ (where $\mathbb{R}^+$ refers to the set of strictly positive real values of $x$). This now fixes the function and allows it to be defined across the entire domain as you should expect from a function.

So the correct way to write this (to avoid points where the function is not defined) is very similar to what you meant by your original question - except we now include this restriction of the domain to give us the function $F: \mathbb{R}^+ \rightarrow \mathbb{R}$ such that:

$$F(x) = \max \{ a^2 \in \mathbb{N} \space : \space a<x, \space a \in \mathbb{N}\}$$