A few problems and continuity and differentiability

In summary: Therefore, $(x + y)^{\frac{1}{3}} < x^{\frac{1}{3}} + y^{\frac{1}{3}}$ for all positive real numbers $x,y$.2. Let $W$ be an open set in $\mathbb{R}^n$. Let $p \in W$ and $q \notin W$. Prove that there is a boundary point of $W$ on the line segment joining $p$ and $q$. Proof: Since $W$ is an open set, there exists a neighborhood $U$ of $p$ such that $U \subseteq W$. Since $q \notin W$, $q \notin U
  • #1
mathhelps
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Homework Statement


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 $(0,0)$.

Homework Equations


The Attempt at a Solution

 
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  • #2
1. Using simple algebra, prove that {\(x+y)^{\frac{1}{3}}<x^{\frac{1}{3}}+y^{\frac{1}{3}}}for x>0,\ y>0. Proof: Let $x, y$ be positive real numbers with $x > 0, y > 0$. Then, \begin{align*} (x + y)^{\frac{1}{3}} &< x^{\frac{1}{3}} + y^{\frac{1}{3}} \\ \Leftrightarrow (x + y)^{\frac{3}{3}} &< (x^{\frac{1}{3}} + y^{\frac{1}{3}})^3 \\ \Leftrightarrow (x + y) &< (x^{\frac{1}{3}} + y^{\frac{1}{3}})^3 \\ \Leftrightarrow x + y &< x + 3x^{\frac{2}{3}}y^{\frac{1}{3}} + 3x^{\frac{1}{3}}y^{\frac{2}{3}} + y \\ \Leftrightarrow 0 &< 2x^{\frac{2}{3}}y^{\frac{1}{3}} + 2x^{\frac{1}{3}}y^{\frac{2}{3}} \\ \Leftrightarrow 0 &< x^{\frac{2}{3}}y^{\frac{1}{3}} + x^{\frac{1}{3}}y^{\frac{2}{3}} \\ \Leftrightarrow 0 &< \dfrac{x^{\frac{2}{3}}y^{\frac{1}{3}} + x^{\frac{1}{3}}y^{\frac{2}{3}}}{x^{\frac{2}{3}} + y^{\frac{2}{3}}} \\ \Leftrightarrow 0 &< \dfrac{x^{\frac{2}{3}} + y^{\frac{2}{3}}}{x^{\frac{2}{3}} + y^{\frac{2}{3}}} \\ \Leftrightarrow 0 &< 1\end{align*
 

1. What is continuity and differentiability?

Continuity and differentiability are concepts in calculus that describe the smoothness of a function. Continuity means that a function is unbroken and has no gaps or holes, while differentiability means that a function has a well-defined slope at every point.

2. How do you determine if a function is continuous?

A function is continuous if the value of the function at a given point is equal to the limit of the function as it approaches that point. In other words, the left and right limits of the function must be equal at that point.

3. What is the difference between continuity and differentiability?

Continuity and differentiability are related concepts, but they have different criteria. A function can be continuous without being differentiable, but a function must be continuous to be differentiable.

4. What are the common types of discontinuities?

The common types of discontinuities are removable, jump, and infinite discontinuities. A removable discontinuity occurs when there is a hole in the graph of a function, a jump discontinuity occurs when there is a sudden jump in the graph, and an infinite discontinuity occurs when a function approaches infinity at a certain point.

5. How can you test for differentiability?

A function is differentiable if it has a well-defined derivative at every point. This can be tested using the limit definition of the derivative or by checking if the function is continuous and has a defined slope at every point. Additionally, if a function is differentiable, it will also be continuous.

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