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First, this question makes a brief comment:

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

I am not sure what this means exactly, and why it is mentioned, though clearly it has something to do with the answer from the solution manual which is different from mine in a systematic way.

Now the actual problem:

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 solution manual shows the following enter image description here

Consider $x=0$. The first number in the decimal expansion is also $0$. Negative numbers just below zero look like $-0.000000...$. Why is $f(0)$ equal to nine in the figure above?

Consider $x=0.1$. $f(0.1)=1$. Numbers just smaller than $0.1$ are of form $0.0999999...$. The first number in the decimal expansion is zero. Considering the initial comment about $1.23999... = 1.24000$ this means $0.09999...=0.1$. Wouldn't this still mean that $f(0.1)=1$ and not $f(0.1)=0$ like in the graph above?

In the figure above, $f(0.1)$ seems to be $0$. Also,

I came up with the following

enter image description here

which for positive x is the same as the solution manual solution except for the endpoints of the intervals.

xoux
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    Real numbers having finitely long decimal representations have two decimal representations. The example given is $1.23\overline{9} = 1.24$. If one is going to work with decimal representations, this ambiguity must be first resolved. The resolution in the solutions manual is to always is the "infinitely many $9$s" of the two. So the real number with representation $0.1$ also has the representation $0.0\overline{9}$ and by the given disambiguation, you must use the latter, which has first decimal digit "$0$". – Eric Towers Nov 26 '21 at 23:05
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    You may find the Wikipedia plot summary for $0.999\ldots$ useful. – Brian Tung Nov 26 '21 at 23:06
  • @EricTowers What about the number $0$? How is the first number in the decimal expansion of $0$ a 9? – xoux Nov 26 '21 at 23:08
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    And everything about $x \leq 0$ in the solutions manual is wrong. – Eric Towers Nov 26 '21 at 23:09
  • Note that this is the same issue as defining the fractional part of a number. Your solution reflects one interpretation while the solution manual reflects the other. I see no way, from what you've shown here, to resolve the conflict. – Brian Moehring Nov 26 '21 at 23:37
  • For the sake of uniqueness one should not allow a representation which finally ends in period 9. – Michael Hoppe Nov 27 '21 at 10:28

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