# On the function $\chi_{\{x \le F(y)\}}(x,y)$ where $F$ is Lipschitz

Let $$F:\mathbb{R} \to \mathbb{R}$$ be a $$L$$-Lipschitz function.

Consider the function $$G(x,y) = \chi_{\{x \le F(y)\}}(x,y),$$ where $$\chi$$ is the indicator function.

• How can I plot this function using MATLAB or Mathematica in the case, for example $$F(y) = y$$?
• Is it true that $$G$$ is Lipschitz continuous (at least with respect to one of the variables?

Follow-up:

1. Is $$G$$ a BV function?
2. What is its distributional derivative?

If $$G$$ is continuous then it's constant because $$G$$ can assume only values $$0$$ and $$1$$.

The set $$E=\left\{(x, y)|x\leq F(y)\right\}$$ is the subgraph of function $$F$$ where $$x$$ axis is changed with $$y$$ axis, and its topological boundary is $$\partial E=\left\{(x, y)|x=F(y)\right\}$$ and coincides with $$F$$ graph. Because $$F$$ is Lipscitz then $$E$$ boundary is also Lipscitz then $$E$$ has locally finite perimeter that implies $$\chi_E(x, y)$$ is a locally bounded variation function and its exterior normal $$\nu_E$$ is well defined on $$\mathcal H^1$$-almost every point over $$\partial E$$.

Also $$\nu_E[F(y), y]=\left(-\frac{1}{\sqrt{1+\left\lvert F'(y)\right\rvert^2}}; \frac{F'(y)}{\sqrt{1+\left\lvert F'(y)\right\rvert^2}}\right)$$ because $$F$$ derivative is defined almost everywhere.

Due to De-Giorgi Structure theorem we have $$D\chi_E(A)=\int_{A\cap\partial E}\nu_E(x, y)d\mathcal H^1$$