1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A few problems and continuity and differentiability

  1. Oct 23, 2011 #1
    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 [itex]{\(x+y)^{\frac{1}{3}}<x^{\frac{1}{3}}+y^{\frac{1}{3}}}[/itex]for [itex]x>0,\ y>0[/itex]. 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
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: A few problems and continuity and differentiability
Loading...