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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...”

3 Answers3

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  1. 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.

  2. 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.

hardmath
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SBF
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  • I think Wikipedia is very reliable, especially when it comes to Mathematics. Yes, one should be cautious but that is true for any source. http://math.stackexchange.com/questions/744132/are-mathematical-articles-on-wikipedia-reliable I think most people agree. – The very fluffy Panda Jul 24 '14 at 12:20
  • @PandaBear: thanks, I think the OP may find this link useful as well. Still, it's just my opinion regarding the subject, and although I do math on Wikipedia, I'd never use the material from there in any way before checking this out in a book. That's true, books also contain mistakes, but the probability is lower. – SBF Jul 24 '14 at 12:49
  • Your definition leaves the right-hand endpoint without a function value. Is this not a problem? –  Jul 25 '14 at 09:54
  • @EsperantoSpeaker1: not really, $1_A$ is defined everywhere, so it their sum, so $f(x_n) = 0$. One may not like it and want $f(x_n) = f_{n-1}$, yet I think this is not the main issue that bothers the OP. – SBF Jul 25 '14 at 10:04
  • "I think it depends on the context." - "Give me the benefit of your convictions, if you have any; but keep your doubts to yourself, for I have enough of my own." -- Goethe –  Jul 26 '14 at 08:37
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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.

MPW
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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.

John Bentin
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  • "more acceptable"? My point is that I believe that the given definition is not at all acceptable. –  Jul 26 '14 at 08:35
  • @EsperantoSpeaker1: Something that is acceptable, even to a small degree, is more acceptable than something that is not at all acceptable (as, we agree, is the Wiktionary definition). – John Bentin Jul 27 '14 at 11:34