On page $276$ of his book Stromberg mentions that $\xi_{P}$ (where $P\subset[0,1]$ is a Cantor type set) can be written as the pointwise limit of a sequence of contiuous functions on $[0,1]$.
He then suggests the following sequence: $f_n=1$ on $P$ and $f_n(x)=0$ if $\text{dist}(x,P)\geq 1/n$.
I assume he means $f_n(x)=1$ if $\text{dist}(x,P)< 1/n$. But then $f_n$ is not continuous at $x$ if $\text{dist}(x,P)=1/n$.
Am I missing something?
I believe that the primary difficulty here is that the author has elided some details. The author wants to approximate the characteristic function of a Cantor set by a sequence of continuous functions, $f_n$. He gives us the important properties of the function, but skips the details: namely,
$f_n(x) = 1$ if $x \in P$ (where $P$ denotes the Cantor set being considered),
$f_n(x) = 0$ if $d(x,P) > 1/n$ (where $d(x,P) = \inf\{ d(x,y) \mid y\in P\}$ is the distance from $x$ to $P$; pardon the overloading of notation), and
$f_n$ is continuous.
It is evident that if such a sequence of functions exists, then it converges pointwise to $\chi_P$ (the characteristic function of $P$):
If $x \in P$ then $f_n(x) = 1$ for all $n$. Thus for $x \in P$, $f_n(x) \to 1 = \chi_P(x)$.
If $x \ne P$ then (because $P$ is closed) $d(x,P) > 0$, which means that there is some $N$ so large that $n > N$ implies that $d(x,P) > 1/n$. Hence $f_n(x) = 0$ for all such $n$. Thus for $x \not\in P$, $f_n(x) \to 0 = \chi_P(x)$.
The only trick is to ensure that each $f_n$ is continuous. There are a lot of ways that this can be done. An "obvious" technique is to use linear interpolation: define $f_n(x) = 1-n d(x,P)$ for any $x$ such that $0 < d(x,P) < 1/n$. It is also possible to build the $f_n$ so that they are smooth (the usual technique would be to convolve $\chi_P$ with a smooth bump function supported on a ball of radius $1/n$, or something like that).
The point is that these functions do not need to be given precisely—it is sufficient to know that there exist functions which get the job done. Because an exact pointwise definition is inessential (and, frankly, is likely to bog the exposition down in irrelevant detail), the author has chosen to omit such definitions. This is pretty standard practice in mathematical writing: leave out the bits that you expect an informed reader to fill in on their own. It is then up to the reader to fill in the details.