For each integer $n$ and $x\in (0,1)$, let $x = 0.k_1k_2k_3...k_nk_{n+1}...$, where $k_i$ is an element of $\{0,...,9\}$ be the decimal expansion of $x$. For such $x$, define $f_n(x) = 0.k_1k_2...k_n$.
a) Show that $f_n$ is measurable.
b) Compute the integral of $f_n$ over $(0,1)$.
c) Compute the limit of the integral of $f_n$ over $(0,1)$ as $n$ approaches infinity, AND compute the integral of $lim f_n$ over $(0,1)$ as $n$ approaches infinity.
b) and c) are (of course) computational, and not so much of a problem - although, I'd appreciate comments on those, just in case it gives interesting counterpoints!
My real question is this: $f_n$ is essentially defined in such a way that it produces a step function on $(0,1)$, is it not? In other words, $f_n$ truncates members from a set, resulting in a countable image, like so: $f_5(x) = 0.k_1k_2k_3k_4k_5$, so if we take an interval such as $(0.123450,0.123459)$, EVERY $x$ in this interval will map to $0.12345$ in $f_5$, and likewise for other intervals. Thus, $f_n$ is the sum of a series of simple (in fact, single-valued, characteristic) functions. As characteristic (therefore continuous) functions, they are measurable, because continuous functions map intervals to intervals, and all intervals have measure. And so $f_n$, as a sum of measurable functions, is also measurable.
(A sidenote: it would appear that each $f_n$ can be represented as the sum of exactly $10n$ characteristic functions, each of which is defined on (again) $10n$ disjoint subintervals of $(0,1)$.)
You are right that $f_n$ is a step function. Indeed, for each fixed $n$, the image of $f_n$ are $\{y = 0.k_1...k_n: k_i \in \{0,1,...,9\}\}$. This is a finite set (not only countable).
On the other hand, for each $y$ in the image, $f^{-1}(y)$ is the interval
$$f^{-1}(y) = [y , y + 10^{-n})$$
(which is not exactly the same as your example $(0.12345, 0.123459)$)
Note that this shows already that $f_n$ is measureable, as for ANY set $A$, $$f^{-1}(A) = \bigcup_{y\in A} f^{-1}(y)$$
is the finite union of half open intervals, thus is measureable.
Note that characteristic function are Not continuous in general.