What is meant by $f(x) = [x]$ on $[0, a]$? This exercise appears in Spivak's Calculus (4th edition, p. 130 exercise 1.11) and I'm unclear on the notation.

Let $f(x) = [x]$ on $[0, a]$. Determine whether $f(x)$ is bounded above or below and/or takes on a maximum or minimum value in the indicated interval.

The solution is:

$f$ has a maximum and minimum value, since $f$ is continuous.

 A: I love Spivak's Calculus!
You're absolutely right that the given function has a maximum and a minimum; however, it is not because it is continuous - the reason is far more elementary:
Suppose $f(x) = [x]$ is defined to be the greatest integer that is less than or equal to $x$; that is, given $x \geq 0$, round down, or equivalently, take its integer part. In that case, if $a = 3$, then for $x \in [0,1)$ $f(x) = 0$, for $x \in [1,2)$ $ f(x) = 1$, and $f(2) = 2$. This demonstrates that $f$, as defined, is discontinuous, and will remain discontinuous so long as $a \geq 1$. 
Similar behavior is exhibited if $f$ is defined as the Ceiling function, which returns the smallest integer that is greater than or equal to the input (for example, Ceil(3.1) = 4), or if $f$ is defined to be the nearest integer function. Note that in the latter case, you need to decide what happens at positive integer multiples of $\frac{1}{2}$, as all such numbers are always equidistant from 2 unique integers. 
Hope this helps!
