Exercise 4.3.6

Question

Provide an example of each or explain why the request is impossible.
  \begin{enumerate}[label=(\alph*)] 
  \item Two functions $f$ and $g$, neither of which is continuous at 0 but such that $f(x) g(x)$ and $f(x)+g(x)$ are continuous at 0
  \item A function $f(x)$ continuous at 0 and $g(x)$ not continuous at 0 such that $f(x)+g(x)$ is continuous at 0
  \item A function $f(x)$ continuous at 0 and $g(x)$ not continuous at 0 such that $f(x) g(x)$ is continuous at 0
  \item A function $f(x)$ not continuous at 0 such that $f(x)+\frac{1}{f(x)}$ is continuous at 0 .
  \item A function $f(x)$ not continuous at 0 such that $[f(x)]^{3}$ is continuous at 0 .
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Let

$$f(x) = \begin{cases}1 &\text{ if } x \ge 0 \ -1 &\text{ if } x < 0\end{cases}$$
    And set $g(x) = -f(x)$. we have $f(x) + g(x) = 0$ which is continuous at zero, and we have $f(x)g(x) = -f(x)^2 = -1$ which is also continuous at zero.
  \item Impossible, since it would imply that $(f + g) - f = g$ is continuous at zero (sum of continuous functions is continuous).
  \item Let $f(x) = 0$, then $f(x)g(x) = 0$ is continuous at zero for any $g(x)$.
  \item
Let
    $$f(x) = \begin{cases}2 &\text{ if } x \ge 0 \ 1/2 &\text{ if } x < 0\end{cases}$$
    Then $f(x) + 1/f(x) = 2.5$ is continuous at zero.

\item Impossible, if $[f(x)]^3$ was continuous at zero then $\left([f(x)]^3\right)^{1/3} = f(x)$ would also be continuous at zero since the composition of continuous functions is continuous
   \end{enumerate}

Exercise 4.3.6

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Exercise 4.3.6

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