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Could someone solve the question and explain me

Let $\omega=(-1,1)\times (-1,1)$ and $f,g:\omega \to \mathbb{R}$ are defined by

$$f(x,y) = \begin{cases} 1, & \text{if $x$ $\le$ 0} \\[2ex] 0, & \text{anywhere else} \end{cases}$$ and

$$g(x,y) = \left\lVert x-y \right\rVert.$$

Determine whether $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, $\frac{\partial g}{\partial x}$, $\frac{\partial g}{\partial y}$ exists weakly and state it if they exist.

Do I need to learn distributions to understand this ?

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  • $\begingroup$ Do you know the definition of weak derivatives? You do not need distributions to solve this (homework?) problem. $\endgroup$
    – gerw
    Jan 12, 2017 at 7:44
  • $\begingroup$ @gerw I do understand the definition. But what I do not understand is what norm should I apply for function g. I thinks it is square intergral norm. Is it right ? $\endgroup$
    – AccGen
    Jan 12, 2017 at 8:00
  • $\begingroup$ I do not know what you mean by "norm apply for function $g$". Why do you need a norm for the definition of weak derivatives? $\endgroup$
    – gerw
    Jan 12, 2017 at 8:09
  • $\begingroup$ @gerw the norm given in the question g(x,y) $\endgroup$
    – AccGen
    Jan 12, 2017 at 10:06
  • $\begingroup$ Oh, I see. Since $x, y$ are just scalars, I would suppose that this is the absolute value of the difference. Isn't it? $\endgroup$
    – gerw
    Jan 12, 2017 at 10:45

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