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Let's say I have a function, say $f(x)$. However, $x$ is not continuous, it can only be integers. So, I only have definitions for $f(1),f(2),f(3)$, etc. If you were to plot out f of all the integers, and connect the dots, it would look like a monotone decreasing function. Can I call this function monotone even though it's not continuous? And is there a formal way of saying "the continuous, monotone function that would be created by connecting the dots of $f(x)$?

Basically, I want to show that $f(x)$ decreases at a decreasing rate. I can show that $f(x+1) < f(x)$ and that $f(\infty) =$ a constant. Can I do this with an integer-based function?

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JandR, is it that you do not care about the function at any non-integer $x$? Is the function defined for all reals or is it defined at only the integers? – Srivatsan Sep 18 '11 at 4:10
@Srivatsan: Hmm.. I mean that in my case, $f$ is really a recursive function, and $x$ is the number of times of recursion. So I have some inner function say $g(y)$ and $f(1) = g(y)$, $f(2) = g(g(y))$, $f(3) = g(g(g(y)))$ etc. I say $f$ is only defined for integer values of $x$ because I don't know how to interpret a partial recursion. Does that make sense? But, I still want to be able to treat $f(x)$ as continuous, or at least make inferences about its rate-of-rate of change and convergence.. – OctaviaQ Sep 18 '11 at 4:14
JandR Functions $f : \mathbb N \to \mathbb R$ are also called sequences since you can identify it with the sequence $f(1), f(2), \ldots$. For such functions, monotonicity, convergence, limits etc. are all well-defined. – Srivatsan Sep 18 '11 at 4:20
@Srivatsan: Oh, ok, thanks. I will go read up on sequences. I hadn't thought of that! – OctaviaQ Sep 18 '11 at 4:24
up vote 2 down vote accepted

You can call the function monotonic, because the integers are a totally ordered set, and hence a partially ordered set. All that is needed for the definition of monotonic to make sense is that the domain and range of the function be partially ordered. Then a monotonic function is one which is either order-preserving or order-reversing. In particular, a function from any subset of the real numbers to any subset of the real numbers can be called monotonoic if it is either order-preserving or order-reversing.

As for talking about extending the function, I would call it the piecewise affine extension of $f$ to the real numbers which is affine on the interval $[n,n+1]$ for each integer $n$.

Finally, if you want to talk about this function (the piecewise affine extension of $f$) decreasing more slowly, you are talking about the concavity of the function. Specifically, you are saying that the function is decreasing and convex. Convexity does not follow even eventually from the fact that it is decreasing and bounded below. Take the example that $f(n)=\frac{1}{\lfloor (n+1)/2 \rfloor}$ for $n>0$ and $f(n)=1$ otherwise.

What is true, however, is that the finite difference (the discrete analogue of a derivative) is always negative and has limit 0. It does not follow that the 2nd finite difference is always positive, it does follow that the second finite difference tends to 0 also, and is positive on average (for some sense of the word average).

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The more usual term is "piecewise linear", even though in principle "piecewise affine" is better. – Robert Israel Sep 18 '11 at 5:29
true. I prefer piecewise affine and always use it, but piecewise linear is basically standard. – Logan Maingi Sep 18 '11 at 5:41

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