This isn't a numerical question but moreso conceptual. So, my Calculus textbook has the following paragraph regarding Local Extrema:

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"If the domain of ƒ is the closed interval [a, b], then ƒ has a local maximum at the endpoint \[x=a\] if \[f(x) \le f(a)\] for all x in some half-open interval Likewise, f has a local maximum at an interior point \[x = c\] if \[f(x) \le f(c)\]for all x in some open interval \[(c -\delta, c+\delta), \delta > 0\] and a local maximum at the endpoint \[x = b\]if \[f(x) \le f(b)\]for all x in some half-open interval \[(b -\delta,b],\delta > 0.\] The inequalities are reversed for local minimum values. In Figure 4.5, the function ƒ has local maxima at c and d and local minima at a, e, and b. Local extrema are also called relative extrema. Some functions can have infinitely many local extrema, even over a finite interval." I get why [a, b] has to be a closed interval, but I don't understand what the consequences would be for the above statements regarding local extrema if they had closed intervals. i.e., why are some of them half-open? I don't understand how them being closed would be an issue or challenge the logic of the statements.

Whoops, missed a portion. Right after "in some half-open interval" for an endpoint maxima for a:\[[a, a+\delta), \delta > 0\]

Is it just because delta can't be actually quantified/defined and is essentially an arbitrary number, while [a,b] are definite fixed end points? So basically the open-ended statement is just a reinforcement that the half-open interval can be anything?

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