If $f$ is Discontinuous and $g$ is Continous, then $f(g(x))$ is Continous

Question

If $f$ is Discontinuous at $x=a$ and $g$ is Continuous, then $f(g(x))$ is Continuous. With nature of these functions we have plenty of examples where $f(g(x)$ is Discontinuous.

But consider the example:

$$f(x)=\begin{cases} 0, \quad & \text{ if } x \ne 0, \\ 1, &\text{ if } x = 0\end{cases}$$ which is Discontinuous at $x=0$ and let $g(x)=0$ Then we have

$$f(g(x))=1$$ which is Continuous.

Are there any other example satisfying this?

Answer 1

Sure. And $g$ doesn't have to be constant. For instance, take $f(x)=\lfloor x\rfloor$ and $g(x)=\frac{x^2}{1+x^2}$. THen $f\bigl(g(x)\bigr)=0$ for every real $x$

Answer 2

Yes take $$f(x)=\lfloor x \rfloor$$ which is Discontinuous at $x=0$ and $$g(x)=x^2$$ which is Continuous at $x=0$ Now

$$f(g(x))=\lfloor x^2 \rfloor $$ is Continuous at $x=0$