Can the composition of a continuous and a discontinuous function be continuous?

Question

Let $I$ be an interval in $\Bbb{R}$ and $f, g : I\to \Bbb{R}$ such that $f$ is continuous and $g$ is discontinuous.

Question: Can either of $f\circ g$ or $g\circ f$ be continuous?

score 3 · Accepted Answer · answered May 18 '19 at 10:13

3

Yes. Take $I=\mathbb{R}$, $f\equiv 0$, $g$ is Dirichlet function. Both $f\circ g$ and $g\circ f$ are continuous in that case.

answered May 18 '19 at 10:13

Mark

You should briefly recall what Dirichlet function is... – Jean Marie May 18 '19 at 12:18
This is a very known function. It is $1_{\mathbb{Q}}$. – Mark May 18 '19 at 12:30
As a teacher, I think that it is always good to recap important things (self contained content) especially when you have room, whence my remark. Moreover, it is a "very well known function" for analysts but not especialy for the "vulgus pecus". – Jean Marie May 18 '19 at 12:59

score 1 · Answer 2 · edited Jun 12 '20 at 10:38

Here is a simple example.

Take

for function $f$ the continuous "absolute value" function,
for function $g$ the discontinuous function (curve represented in red on the figure below) defined by :

$$g(x)=\begin{cases}-x-1&\text{if} \ -1 \leq x < 0\\ -x+1&\text{if} \ \ \ 0 \leq x \leq 1\\ 0&\text{elsewhere}\\ \end{cases} ;$$

$g$ is a discontinuous function .

Then

$$(f \circ g)(x)=(g \circ f)(x)=t(x)$$

where $t$ is the continuous so-called "tent" function (blue curve on the figure).

2 Answers2