1.
Write two or more paragraphs describing the best ways to get stuck and unstuck on a math problem.
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Worksheet 3.5 Homework 05
Instructions: Complete all the exercises below and submit your work via Gradescope by Monday March 2 at 10pm.
2.
Let \(f,g\colon \mathbb R^n\to \mathbb R\) be two continuous maps. Suppose there is a number \(N\) such that \(\lVert x\rVert \gt N\) implies \(g(x)\le 0\) and \(f(x)\ge 1\text{.}\) Prove that the subspaces
\begin{equation*}
\{x\in \mathbb R^n\mid f(x)=g(x)\}\quad\text{and}\quad \{x\in \mathbb R^n\mid f(x)\le g(x)\}
\end{equation*}
are compact.
3.
For \(X\) a space, the cone on \(X\) is
\begin{equation*}
CX := ([0,1]\times X)/{\sim}
\end{equation*}
where \((s,x)\sim (t,y)\) means \((s,x)=(t,y)\) or \(s=t=0\text{.}\)
(a)
Draw a picture of \(CS^1\text{.}\)
(b)
Prove that the map \(f\colon CS^n\to D^{n+1}\) given by \([(t,x)]\mapsto tx\) is well-defined and continuous.
(c)
Use the Fundamental Theorem of Topology to prove that \(f\colon CS^n\to D^{n+1}\) is a homeomorphism.
4.
Let \(n\gt 0\text{.}\) Prove that there is a homeomorphism \(D^n/{\sim}\cong S^n\) where \(x\sim y\) means \(x=y\) or \(x,y\in S^{n-1}\subseteq D^n\text{.}\)
5.
Prove that
\begin{align*}
s\colon [0,\infty) \amp \longrightarrow \mathbb R\\
t \amp \longmapsto (t-\lfloor t\rfloor)^{\lceil t\rceil}+\lfloor t\rfloor
\end{align*}
is continuous. Here \(\lfloor~\rfloor\) and \(\lceil~\rceil\) are the floor and ceiling functions, respectively. (Instead of proving this statement directly, you might enjoy developing a lemma about when piecewise defined functions are continuous, and then applying that lemma in this case.)
