Is this function $f(x,y)$ regulated when fixing $x$ and then when fixing $y$?

Question

I have come across this question when studying for my exams. I have gotten somewhere but I am struggling coming up with solid reasons on whether it is regulated or not.

The function is $f:[0,1]^2 \to [0,1]$ defined by

$$ f(x,y) = \begin{cases} \frac{1}{y^2} &\text{if } 0 < x< y < 1\\ -\frac{1}{x^2}&\text{if }0 < y < x < 1\\ 0 &\text{otherwise}\\ \end{cases}$$

(a) For fixed y is $ x \to f(x,y)$ a regulated function? Calculate $\int_0^1 f(x,y) \,dx$.

So far I have that $ y \in (0,1) $, so taking $y = \frac{1}{2}$ for instance and drawing this function I can see that it is not monotonic. Hence, I can't conclude that its regulated via this method. I tried constructing a sequence of step functions and showing they converge to prove its regulated via this way but I can't manage to do this.

For the integral I have:

$$ \int_0^1 f(x,y) \,dx = \int_0^\frac{1}{y} \frac{1}{y^2} \,dx + \int_\frac{1}{y}^1 -\frac{1}{x^2} \, dx = \frac{1}{y^3} - y + 1$$

I can draw the function when $f(x,y)$ has a fixed $y$ however I can't manage to draw it when $x$ is fixed which is part (b) of this question.

(b) For fixed $x$ is $ y \to f(x,y)$ a regulated function? Calculate $\int_0^1 f(x,y) \,dy$.

For this part I am having no luck at all.

Overall, my question is does my approach in answering this problem seem correct? How can I conclude if the functions in (a) and (b) are regulated?

ARE they regulated? I am still unsure.

Answer 1

1. For $y \in (0,1)$, $F_y(x)=f(x,y)$ is continuous on $(0,y)$ and $(y,1)$. As all the limits of $F_y(x)$ at $0^+,y^-,y^+,1^-$ exist, $F_y(x)$ is regulated.

The bounds of your integrals are not correct.

2. $F_x(y)$ is regulated for similar arguments.