Which Definition Should Be Used for Relative Maximum

Question

1. Look at the graph of $f$:

2. Below is the definition of a relative maximum that I am using:

   Definition: Let $f:X\rightarrow Y$ be a function. We say $c\in X$ is a relative maximum of $f(x)$ if $\exists(a, b)\subseteq X$: $a<c<b$ and $\forall x\in (a, b)$ $f(x)\leq f(c)$.

Looking at the answer found in $2.$ a neighborhood containing $-3$ can be any set $N$ satisfying: $[-3, \delta-3)\subseteq N \subseteq X$ for some $\delta>0$. Therefore, having $N=[-3, \delta-3)$ where $\delta=1$ would clearly result in $f(-3)$ being called a "relative maximum."

This is a problem because my definition does not state $-3$ would be a relative maximum as there is no open interval containing $-3$ here. The domain of $f$ cannot stretch out that far.

Question: If one were teaching Calculus, what definition is better to use? Is $f(x)$ a relative maximum at $x=-3$ here or not...

Answer 1

Instead of requiring $(a,b)\subset X$ as in your definition, the usual definition would instead require that $f(x)\leq f(c)$ for all $x \in X \cap (a,b)$ (where $a < c < b$).

With this (usual) definition, your solution is correct.

With the (unusual) definition that you cite, the point -3 would not be a local (or relative) maximum.

Answer 2

For teaching, you ought use the definition used in the text. Since this definition usually only applies to points in the interior of the domain [as a subset of $\mathbb{R}$], we should not call the point at $x=-3$ a relative max in the classroom.

That said, in almost every discussion I have had with other teachers at the college level, we concluded the appropriate definition for boundary points is via the subspace topology, which means it ought to be a "relative max" in the generalized definition.