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In my real analysis text book there is a question that says:

Decide whether $f(x)=[x]$ is bounded above or below on the interval $[0,a]$ where $a$ is arbitrary, and whether the function takes on it's maximum or minimum value within that same interval.

This question is very straightforward, assuming $[x]=x$. But if that is the case, then the choice of notation is very strange.

Is there another way to interpret the notation's meaning?

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    The question is from Michael Spivak's "Calculus" - 4th Edition (also 3rd edition). It's part (xii) of chapter 7 question 1.2011-05-01

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It had been fairly standard for $[x]$ to represent "the greatest integer not greater than $x$" (aka, the "floor" function). With fancier type-setting options allowing for $\lfloor x \rfloor$ for a more suggestive "floor" notation ---as well as $\lceil x \rceil$ for the counterpart "ceiling" ("smallest integer not smaller than $x$")--- I've seen $[x]$ taking on the role of "nearest integer" (that is, the "rounding" function) although $\lfloor x \rceil$ is also available for this, freeing up $[x]$ for author's discretion.

With regard to @Brandon's "fractional part", I've seen that more often represented as $\{ x \}$, usually in conjunction with the floor interpretation of $[x]$, so that one would write $x = [x] + \{x\}$ (at least for non-negative $x$).