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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