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The question: “If a function is not monotone on $(a, b)$, then its square cannot be monotone on $(a, b)$.” We are to provide a counterexample to this statement.

On initial attempts I was able to forge piecewise functions which were sufficient counterexamples to this statement. One is stated below.

The function: $(-1)^{\lfloor x\rfloor}x$. (we can consider the required domain). Here's a graph I made on desmos

graph of the function

After some attempts I was able to find another function.

The function: $(-1)^xx$. (we can consider the required domain). Graphed here

graph of the function

Here is where I have a doubt; is this function even a sufficient counter example for the statement?

My conclusion against this function being sufficient came in the following way:

  1. For a function to be monotonic increasing in $(a, b)$ we must have: $f(x_2) > f(x_1)$ for $x_2 > x_1$; if $x_1 , x_2\in(a, b)$. (vice versa for decreasing)

  2. Now the function stated above, returns complex numbers for some values in any interval we consider; now by property of complex numbers we know we can’t compare complex numbers so this is technically breaking the rigorous definition of monotonicity…

  3. On the other hand it is impossible to modify the domain in such a way that the output to them lies only on the real numbers, as there are infinitely many such points which would yield a complex number between two points

So my question is, is my train of thought against the latter function correct or am I missing any fine detail, in the analysis of its behavior.

Thank you in advance. Any insight provided will be invaluable to drive my curiosity in Mathematics.

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    $\begingroup$ Proofread, edit, MathJax. $\endgroup$ Commented Oct 29 at 17:17
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    $\begingroup$ It is true for continuous functions. $\endgroup$ Commented Oct 29 at 17:18
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    $\begingroup$ It is also true for non-negative functions. $\endgroup$ Commented Oct 29 at 17:30
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    $\begingroup$ What exactly do you mean by "not defined piecewise"? Does something like $(-1)^{\lfloor x \rfloor} x$ count? $\endgroup$ Commented Oct 29 at 17:35
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    $\begingroup$ What does $(-1)^x$ mean? $\endgroup$ Commented Oct 29 at 17:46

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Your definition of piece-wise is a bit ambiguous. As noted, this fact is true for both continuous, non-negative and non-positive functions. Here's a function that I feel under any reasonable definition of 'non-piecewise' that does not imply continuity should be non-piecewise. Define $f:\mathbb R\to\mathbb R$ by

$$f(x)=\lim_{y\to x^+}\frac{|y^2-1|}{y^2-1}$$ This function takes values in $\{-1,1\}$ and is not monotone (for instance $f(-2)=1$, $f(0)=-1$, $f(2)=1$) but its square is monotone, as it's constant $1$

Here's a drawing I made with geogebra of $f$, together with $f(-1)=-1$ and $f(1)=1$

I also think you shouldn't be worried about what a 'piecewise function' is. It's not a well defined term, and they give just as easy (if not easier) counterexamples to the claim.

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  • $\begingroup$ I understand the disambiguation around the definition,and I've made the necessary edits. I will be very glad if you could provide some kind of exposure on the latter function I mentioned: (-1)^x.x. My primary concern revolves around the synopsis of this function and the validity of my arguments as I mentioned in the question. $\endgroup$ Commented Oct 30 at 5:38
  • $\begingroup$ @AbhinabaChatterjee I don't think it's a sufficient counterexample. But it only boils down to definitions. Is a non monotone function by definition a real function? Is the codomain of $f$ assumed to be real? If the answer to both of those questions is no, then it is a valid counterexample. Otherwise, it's not $\endgroup$ Commented Oct 30 at 14:43
  • $\begingroup$ @AbhinabaChatterjee asides from the clear fact that $(-1)^x$ is not a well defined function, you would have to choose a branch for every $x$ that is not a rational number with odd denominator $\endgroup$ Commented Oct 30 at 14:44
  • $\begingroup$ I am truly grateful. However being not very versed in this topic I have a few more questions: 1. What is the proper "legal" textbook definition of a non monotone function(if any). 2. How can we assume the codomain of f to be real? What about the complex outputs it generates? Do we ignore them? Also from what has been taught to me, we can only curtail the domain of a function and can not interfere with it's range. And in the case of this specific function is it possible to define a domain where the function will output only real values? 3. What do we mean by well defined? $\endgroup$ Commented Oct 30 at 17:00
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Take any positive monotone function and build a new function by taking the square root and assigning varying signs.

E.g. $e^x$ vs. $\text{sign}(\sin(x))e^{x/2}$.

enter image description here

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  • $\begingroup$ But my question still stands about the validity of the function (-1)^x.x, is this function a sufficient counter example? $\endgroup$ Commented Oct 30 at 5:40
  • $\begingroup$ @AbhinabaChatterjee: $x^2$ is not monotonic. $\endgroup$ Commented Oct 30 at 9:10

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