Sampling and reconstruction

From a previous exam that I've been trying to solve for a while there is one section with a series of yes/no questions that I just can't figure out for certain what they should be.

**1) The spectrum of a box filter (in time domain) is also a box function (in frequency domain).**

I've been told this one is false, but why is it false? Also I'm confused by the terminology, what is the difference (if any) between a box filter and a box function?

I know what a box function is -- any function that is zero over the entire real line except for a small interval where it is equal to the constant A, i.e. the "Boxcar function".

If I assume as premise for the following reasoning that a box filter is just a type of box function -- used as a type of filter -- then:

A box filter given a signal in time domain will parse this signal giving only values of 0 and A -- depending on the signal given and the nature of the box function.

Now, the frequency domain is a frequency graph where the quantified distribution of a specific signal's frequency spectrum is shown.

I.e. the frequency spectrum resulting from the former function is simply 0 and A -- why is this not a box function in frequency domain?