I have the (discontinuous) function that reports a $0$ if $X>(\frac{2y}{ln(y)})$ and a $1$ if $X \leq (\frac{2y}{ln(y)})$. I would simply like to describe something like the intuition that this function is "increasing" in $Y$ and "decreasing" in $X$.
For a smooth continuous function I would do this by reporting the sign of first derivative. But a unit step function that outputs $0$ until some threshold and then (discontinuously) outputs $1$ isn't doesn't have a signed first derivative at any point. Nevertheless it seems pretty intuitive to me that a function that reports '$0$ if $Y<0$, $1$ if $Y>0$' is "increasing" in $Y$. That is, an increase in $Y$ is the only thing that increases the function's output value from $0$ to $1$.
And so the function I reported at the beginning, using L'Hospital's rule and a little algebra, seems in the same sense to be "increasing" in $Y$ and "decreasing" in $X$. I believe that this can be handled with a dirac delta function, but I'm not overly familiar with them. And all I'd really like to do is (mathematically) make clear my (our?) intuition that, if the function reports $0$ then as $Y$ increases it becomes 'more likely' for the step function to return a positive value.
Any advice or help on this is greatly appreciated!
A function $f(x)$ is increasing (respectively, decreasing) in the variable $x$ if: $$\text{whenever } x_1<x_2, \text{ we have } f(x_1)\leq f(x_2) \text{ (respectively, } f(x_1)\geq f(x_2)\text{)}.$$ For a multivariate function, say $f(x,y)$, you would alter the definition to say that $f$ is increasing in $x$ if each fixed $y$, $x_1<x_2$ implies $f(x_1) \leq f(x_2)$. Similarly for decreasing. By this definition, your function is increasing in $y$ and decreasing in $x$.