Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak derivative?

share|improve this question
    
What have you tried? –  Davide Giraudo Oct 29 '12 at 10:46
2  
What exactly do you mean by "weakly differentiable" here? There are several possible definitions in different contexts. Do you want the weak derivative to be a function, a distribution, ...? –  Nate Eldredge Oct 29 '12 at 12:40

1 Answer 1

We see that $ sgn(x) = 2H(x) -1 $ where $ H $ is the Heaviside function with $ H(0) = \frac{1}{2}$. Then distribution derivative of $ sgn $ would be $ 2\delta_x $ which is not induced by any function. So $f$ is not weakly differentiable.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.