Is there more than one meaning of the notation "$f(x)=[x]$"? 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?
 A: 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 $x$).
A: Late to the party, I know ...
I've gone back to my own copy of Spivak's "Calculus" (I seem to have the 1967 edition) and looked it up in the Glossary of Symbols at the back, and there it is on page 573, referring me back to its definition on page 70.
It is indeed defined, in Problem 15 of Part 2 Foundations, chapter 4 Graphs, as the greatest integer $\le x$.
Beware, though, because he makes the egregious howler of presenting as an example:
$$[-0.9] = [-1.2] = -1$$
which is particularly bad because this is precisely the mistake that beginners are likely to be confused over and need clarification on.
Of course $[-1.2] = -2$ as $-1 > -1.2$. The greatest integer smaller than $-1.2$ is $-2$.
A: It's certainly not $[x]=x$. 
Rather,  it's undoubtedly the greatest integer function, aka floor function (also denoted $\lfloor x\rfloor$);  that is, the greatest integer less than or equal to $x$.  The ceiling function ($\lceil x\rceil$) is, correspondingly, the smallest integer greater than or equal to $x$..  
According to this, Spanier and Oldham called it the "integer value" in $1987$.  (Incidentally,  I took a course from Spanier, and I consider him to have been an outstanding mathematician. 
Of course,  that puts me in a fairly large club.  But I digress.)
The formula would be:  $\lfloor x\rfloor =n$, where $n$ is the integer such that $n\le x\lt n+1$. 
Another way of describing it, would be to take the number's decimal representation, and truncate it.  That is, set the part after the decimal to zero. 
