\documentclass[12pt]{article}
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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\v{\boldsymbol{v}}

\begin{document}

\title{Analysis II}
\author{Homework 7}
\date{Due in class Monday, April 1, 2019}
\maketitle

\begin{enumerate}


\item Suppose $f \colon \R^n \to \R$ is continuous and satisfies
\[
  f(x+y) = f(x) + f(y)
\]
for all $x,y \in \R^n$.  Show that $f$ is linear, i.e., that
\[
  f(\lambda x) = \lambda \, f(x)
\]
for all $x \in \R^n$, $\lambda \in \R$.


\item Let $V$ be a finite-dimensional normed vector space.  Recall
from class the \emph{general linear group}
\[
  \operatorname{GL}(V) =
  \{ A \in L(V) \colon A \text{ is invertible} \}
\]
Show that the map $\operatorname{inv} \colon \operatorname{GL}(V) \to
L(V)$ defined by
\[
  \operatorname{inv}(A) = A^{-1}
\]
is differentiable with
\[
  \operatorname{inv}'(A) B = - A^{-1} B A^{-1}
\]

\item Let $E \subset \R^n$ be open and $f\colon E \to \R$ possesses
partial derivatives $\partial_1f,\dots,\partial_n f$ that are bounded
on $E$.  Show that $f$ is continuous on $E$.

\item \emph{Disconcerting Example 1.} Consider the function $f \colon
\R^2 \to \R$ defined by
\[
  f(x,y) =
  \begin{dcases}
    \frac{x^3}{x^2+y^2} & \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}
\]
\begin{enumerate}
\item Compute the directional derivative $D_{\v}f (0,0)$ for every
$\v=(a,b) \in \R^2$.  Is $\v \mapsto D_{\v}f (0,0)$ linear?
\item Show that $f$ is not differentiable at the origin.
\end{enumerate}

\item \emph{Disconcerting Example 2.} Consider the function $f \colon
\R^2 \to \R$ defined by
\[
  f(x,y) =
  \begin{dcases}
    \frac{x^3\,y}{x^6+y^2} & \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}
\]
\begin{enumerate}
\item Compute the directional derivative $D_{\v}f (0,0)$ for every
$\v=(a,b) \in \R^2$.  Is $\v \mapsto D_{\v}f (0,0)$ linear?
\item Show that $f$ is not continuous at the origin.
\end{enumerate}

\item \emph{Disconcerting Example 3.} Consider the function $f \colon
\R^2 \to \R$ defined by
\[
  f(x,y) =
  \begin{dcases}
    \frac{x^2\,y}{x^4+y^2} \, \sqrt{x^2+y^2} &
    \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}
\]
\begin{enumerate}
\item Compute the directional derivative $D_{\v}f (0,0)$ for every
$\v=(a,b) \in \R^2$.  Is $\v \mapsto D_{\v}f (0,0)$ linear?
\item Show that $f$ is not differentiable at the origin.
\end{enumerate}



\end{enumerate}

\end{document}
