Is not Wiktionary’s definition of “step function” incorrect? Wiktionary says that a step function is, “A function from the real line to a finite subset of the real line”. I realize that there is some variation on how things are defined, but this seems too much, since one of the main uses of step functions is to aid in introducing the Riemann integral, but the well-known “salt-and-pepper” function has a finite range, but is not Riemann-integrable. So, my question is: Did Wiktionary just slip up real bad here?
Here is the address of the Wiktionary definition:
http://en.wiktionary.org/wiki/step_function
The definition given by Wiktionary strikes me as the kind of off-hand guess that a freshman in a Calculus class might give to the professor’s query, “Would anyone like to give us the definition of a step function?” - and, after receiving the Wiktionary-reply, the professor replies with, “Well, that’s a nice guess, and I can see where you’re coming from, however...”
 A: *

*Don't ever rely on Wiktionary/Wikipedia regarding maths. At least. I've used to see mistakes (and rather crucial ones) here and there in Wikipedia's articles on math topics. These sources are may be more reliable 1, 2 and 3. Still, if you read some material on any of those websites it's much more safe if you then take a look in a textbook. Better a yellow one from Springer.

*Step functions may be defined depending on the context. Often this is done for the purposes of integration. For example, for Riemann integration a step function is a one which is of the form
$$
  f(x) = \sum_{i=0}^{n-1} f_i \cdot 1_{[x_i,x_i+1)}(x) \tag{1}
$$
and $(x_0,\dots,x_n)$ is partition of some given interval. For Lebesgue integration one is fine even if instead of $[x_i,x_{i+1})$ there is just some measurable set. In fact, I would call the latter functions as simple functions, and use the term step function for those of the form $(1)$ only. For example, a characteristic function of rationals $1_{\Bbb Q}$ which you seem to refer to as a salt-and-pepper function in such case is simple but not step. Overall, I do not think that the term "step function" is formal and commonly agreed upon. I think it depends on the context.
A: This is what Rudin would call a simple function. If the community agrees with a better definition, you could always edit the Wikipedia entry. Just because something is stated on Wikipedia doesn't make it correct, BTW. It is full of inaccuracies.
A: A more acceptable definition might be "something that looks like a histogram". More formally, let $U$ be a real interval which is the union of a countable number of intervals $U_i$ ($i\in\Bbb Z$), each either being empty or having a nonempty interior, such that $x<y$ whenever $x\in U_i$, $y\in U_j$, and $i<j$. Then a function $f:U\rightarrow\Bbb R$ such that $f(x)=u_i$ when $x\in U_i$ ($i\in\Bbb Z$), for some $\{u_i:i\in\Bbb Z\}$, is called a step function.
Added: If we define the the real function $f:\Bbb R\rightarrow\{0,1\}$ by $f(x)=1$ if $x$ is normal, with $f(x)=0$ otherwise, then $f$ satisfies the Wictionary definition of step function.
