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

Question

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?

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Follow-up:

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

Answer 1

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 $$ D\chi_E(A)=\int_{A\cap\partial E}\nu_E(x, y)d\mathcal H^1 $$