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f has no local maximum or minimum, but 2 and 4 are critical numbers.

my professor drew a constant function (horizontal line) from x=1 to x=5. How come there is no local maximum or minimum for a horizontal line? My professor said that all the values of a horizontal line are the absolute maxima and minima...

According to my book, "A function f has a local maximum at c if f(c) >= f(x) when x is near c. [This means that f(c) >= f(x) for all x in some open interval containing c.]" Let's say the open interval is (1,5), and since f(x)=f(c) (because we have a horizontal line), shouldn't f have a local maximum????