In the given solution of this Problem in Spivak's "Calculus", 3rd ed., there are some details, which I fail to comprehend. I think that in order to be clear I have to include two images.
There is a short preliminary text on pg. 73., the last part of which reads as follows:
There is one ambiguity about infinite decimals which must be eliminated: Every decimal ending in a string of $9$'s is equal to another ending in a string of $0$'s (e.g., $1.23999...=1.24000...$). We will always use the one ending in $9$'s.
The problem reads as follows:
19. Describe as best you can the graphs of the following functions (a complete picture is usually out of the question). (i) $f(x)=$ the 1st number in the decimal expansion of $x$.
The following are the given solution and my own handwritten solution:
(The dots mean that these ends of the intervals are "closed" and the arrows mean that these ends of the intervals are "open".)
I agree with the part of Spivak's solution which is to the right of the vertical axis. Note that $f(0.2)=1$ because in the preliminary text it is made clear that $0.2000...=0.1999...$. (To be completely rigorous, shouldn't he replace $1$ on the horizontal axis with $0.999...$?)
However, I don't understand the indicated Intervals to the left of the vertical axis in Spivak's solution. Isn't it rather the case that for example $f(-0.1)=0$ because $-0.1000...=-0.0999...$ like I indicated in my solution? Am I missing something about negative real numbers? Technically $0=0.000...$, so is there a way to express $0$ with another number ending in $9$'s?
The labelling on the horizontal axis merely identifies numbers. There's no need to choose to identify them in a way that matches the particular representation he's using in defining $f$. What if he also defined a second function, $g$, using the ALTERNATIVE representation of finite-decimals, and asked you to draw $f + g$? What labels would you then have him use on the $x$-axis?
For $x = 0$, his rule about $9$s doesn't apply, for there's no decimal ending in a (nonempty) string of $9$s that's equal to zero. So the first digit in the decimal expansion of $0$ is certainly $0$. The indicated solution seems to suggest that the first digit is $10$, which makes no sense at all. (Indeed, in general, it's tough to know what the first digit in a decimal expansion means, unless it's very carefully defined. For instance, is $0.11\ldots$ or $.11\ldots$ the decimal expansion of $1/9$? The first starts with $0$, the second starts with $1$.
If you say "the first nonzero digit," then there's no answer for $0$.
A typical number between $-0.1$ and $0$ is something like $-0.0734$; I guess that one might say that this starts with $0$ (but not 10 ... that's crazy!). So the first dot-line-arrow shape to the left of the $y$-axis in the solutions manual is just plain wrong. What about the second one? A typical number in there is $-0.1302938\ldots$, where we'd have to say that the first digit is either $0$ (probably not what's intended) or $1$, but certainly not $0$.
So ... the solution-manual answer is wrong to the left of the $y$-axis.