I want to prove that, the set: $S=\{(x,y)\in \mathbb R^2\,\,|\,\,0\lt x\lt y \,\,,\,\,\,\,x^x=y^y \}$ $\,\,$ is uncountable.
My idea is the following:
Consider the function $f(x)=x^x\,,\,\,\,x\gt0$. Then:
$$f'(x)=x^x(\ln(x)+1).$$
$f$ is decreasing in the interval $(0,\frac{1}{\mathbb e})$ and increasing for $x\in [\frac{1}{\mathbb e},1]$. And:
$$\lim_{x\to 0}f(x)=1$$
because of $f''(\frac{1}{\mathbb e})\gt0$,$\,\,$ the point: $x=\frac{1}{\mathbb e}$ is a relative minimum value for $f$. Thus, for every $x\in (0,\frac{1}{\mathbb e})$, we can find $y\in (\frac{1}{\mathbb e},1)$ such that, $x^x=y^y$ and $x\lt y$. So $S$ is uncountable.
Is there another proof?
Thanks in advance…
Your proof is perfectly fine. It seems to me that this approach is what the asker had in mind.