Exercise 2.1.4

Question

Give an example of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ such that
\begin{enumerate}[label=(\alph*)]
\item $f$ is not continuous, but $g$ and $g \circ f$ are continuous;
\item $g$ is not continuous, but $f$ and $g \circ f$ are continuous;
\item $f$ and $g$ are not continuous, but $g \circ f$ is continuous.
\end{enumerate}

Prakash Anand · Accepted Answer

For (a) consider $f(x)=\frac{1}{x}$, $\forall x 
eq 0$ and $f(0)=1$. And let $g(x)=1 \forall x$. Certainly $f$ is not continuous, and $g$ is continuous, and also $g \circ f=1$ which is continuous as well.\par

For (b) consider $f(x)=1 \forall x$ and $g(x)=\frac{1}{x}$, $\forall x 
eq 0$ and $g(0)=1$. Certainly $f$ is continuous, and $g$ is not. Also $g \circ f=1$ which is continuous.\par

For (c) consider $f(x)=g(x)=\frac{1}{x}$, $\forall x 
eq 0$ and $f(0)=g(0)=0$. Then $g \circ f=x \forall x$ which is continuous.

Exercise 2.1.4

Answers

Comments

Exercise 2.1.4

Answers

Comments

Add answer