# A few problems and continuity and differentiability

1. Oct 23, 2011

### mathhelps

1. The problem statement, all variables and given/known data
I have an upcoming math test, and these are from the sample exam. I'll post my solutions as I go along. I've submitted this post as is and am going to edit in my attempts. A few of these are "verify my proof is rigorous" others are "i've no idea what i'm doing

1 Using simple algebra, prove that ${\(x+y)^{\frac{1}{3}}<x^{\frac{1}{3}}+y^{\frac{1}{3}}}$for $x>0,\ y>0$. Then prove that [itex[f(x)= x^{\frac{1}{3}}}[/itex] is uniforml\ y continuous on ${\ds (0,\infty)}$.
2. Let $W$ be an open set in \mathbb{R}^n$. Let$\ds p\in W$and$q \notin W$. Prove that there is a boundary point of$W$on the line segment joining$p$and$q$. 3. Suppose that$\ds f'(x)$exists on$(a,b)$and$\ds f'(x) \ne 0$on$(a,b)$. Prove that either$f'(x)>0$for all$x\in (a,b)$or$f'(x)<0$for all$x \in (a,b)$. You may not assume that$f'$is continuous. Hint: prove that$f$is one-to-one. Let$f(x,y)$denote the function defined by the following rules:$f(x,y)=\begin{cases} 0, \text{ if } (x,y)=(0,0)\\ \frac{xy}{\sqrt{x^2+y^2}}, \text{ if } (x,y)\ne (0,0)$Prove that$f$is continuous at all points and has partial derivatives at all points, but is not differentiable at$(0,0)\$.

2. Relevant equations

3. The attempt at a solution

Last edited: Oct 23, 2011