# Problem Set 4

1. Easy
Find the limit, if it exists, or prove that the limit does not exist
1. $$\displaystyle \lim_{(x,y)\to(0,0)} \left(5x^3 - x^2y^2\right)$$
2. $$\displaystyle \lim_{(x,y)\to (0,0)} \frac{xy}{\sqrt{x^{2}+y^{2}}}$$ [Medium]
3. $$\displaystyle \lim_{(x,y)\to (0,0)} \frac{x^{4}-4y^{2}}{x^{2}+2y^{2}}$$.
4. $$\displaystyle \lim_{(x,y)\to(0,0)} \frac{x^2+y}{\sqrt{x^2+y^2}}$$
5. $$\displaystyle \lim_{(x,y)\to (0,0)} \frac{x^2 \sin^2(y)}{x^2+2y^2}$$. [Medium]

2. Easy
Find an example of a continuous function $$f: \mathbb{R}^{n}\to\mathbb{R}^{m}$$ and an open set $$U\subseteq \mathbb{R}^{n}$$ such that $$f(U)$$ is not open. Still assuming that $$f$$ is continuous, suppose there exists a continuous $$g:\mathbb{R}^{m}\to\mathbb{R}^{n}$$ such that $$f(g(x)) = x$$ and $$g(f(x)) = x$$. If $$U$$ is open, must $$f(U)$$ be open? Must $$g(U)$$ be open?

3. Medium
Define a function $$f:\mathbb R^2\setminus\{(x,y): x=0\} \to \mathbb R$$ as follows: $f(x,y) = \frac{\sin(xy)}{x}$ How should you define $$f(x,y)$$ at $$x = 0$$ so that $$f(x,y)$$ extends to a continuous function on all of $$\mathbb{R}^{2}$$?

4. Medium
Prove the following are equivalent:
1. $$f: \mathbb{R}^{m}\to \mathbb{R}^{n}$$ is $$\epsilon$$-$$\delta$$ continuous
2. For every $$V\subseteq \mathbb{R}^{n}$$ open, $$f^{-1}(V)$$ is open
3. For every $$V\subseteq \mathbb{R}^{n}$$ open, $$\forall\,x\in\mathbb{R}^{m}$$, if $$f(x) \in V$$ then there exists an open set $$U\subseteq\mathbb{R}^{m}$$ containing $$x$$ such that $$f(U) \subseteq V$$.

5. Hard
Let $$f,g: \mathbb{R}^{n}\to\mathbb{R}^{k}$$ be continuous functions and suppose that $$D\subseteq\mathbb{R}^{n}$$ is a dense set. If $$f(x) = g(x)$$ for every $$x \in D$$, then $$f(x) = g(x)$$ for every $$x \in \mathbb{R}^{n}$$.

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