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I'm currently learning functions and in my homework there's the problem:

Let $f\colon\mathbb{N}\to\mathbb{R}$ be a function. Come up with a nickname for this type of function and come up with a nickname for the elements in its domain.

I asked my teacher for clarification and he said that we're supposed to think about the characteristics of such functions and describe them using words in English, be creative and think logically.

BroPro
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  • What a weird question. But Jose Carlos Santos is correct. Such a function is called a "real sequence" and as the Domain can be listed one after another then values can be written out one after another $f(1), f(2), f(3),.....$ and if we use the standard notation $f(n)=a_n$ then we can write it as $a_1,a_2,a_3....$. .... but it never would have occurred to me that this is what the teacher was going for. ... Maybe it the question was: "There is a name for these types of function and we've used it many times already. What do you think it is" I'd have had better luck. – fleablood Mar 30 '21 at 16:42
  • "A function which maps discrete values to continuous values" But it doesn't map to continiuous values. If $f(n) = \sqrt{n}$ say there is now meaning of the word "continuous" that would make the sentence "$\sqrt 5$ is a continuous value" make any sense. – fleablood Mar 30 '21 at 16:44
  • Thinking of such functions, I could have the image of a histogram in my head. The horizontal axis is composed of discrete elements, and the height may take any value in $\mathbb{R}$. If your teacher is super-eccentric, I would make the bars very thin, and call the object a "comb" (and the elements of the domain could be referred to as "teeth"). That's probably pushing it, but you said "creative"... – user3733558 Mar 30 '21 at 16:55
  • Personally, I would have merely said: "A function which maps positive integers to real numbers." However, after reading the answer of Jose Carlos Santos, I also agree with it. – user2661923 Mar 30 '21 at 17:57

2 Answers2

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A function from $\Bbb N$ into $\Bbb R$ is a sequence of elements of $\Bbb R$. In this context, the elements of $\Bbb N$ are the indices.

José Carlos Santos
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Think about, say, a function of the form $x(n)=\sqrt{2}/n$ for $n\in\mathbb{N}$. This is then a function from $\mathbb{N}$ to $\mathbb{R}$, and should have a familiar name.

Rough_Manifolds
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