I was wondering if a function $f:[a,b]\rightarrow[c,d]$ is continuous, $g:[c,d]\rightarrow\mathbb{R}$ is continuous, does it necessarily imply that $g\circ f$ is continuous? Are there counterexamples? What is the necessary and sufficient condition for $g\circ f$ to be continuous?
This is not HWQ. I am just wondering if that is possible.
Best Answer
With the sequence definition of continuity it is obvious that $g\circ f$ is continous, because $$\lim_{n\rightarrow \infty} g(f(x_n))=g(\lim_{n\rightarrow \infty} f(x_n)) = g(f(\lim_{n\rightarrow \infty} x_n))$$ because $f$ and $g$ are continuous.
It is hard to say what is necessary that the composition of function is continuous, taking
$$D(x)=\left\{ \begin{array}{rl} 0 & x\in \mathbb{R}\setminus \mathbb{Q}\\ 1 & x \in \mathbb{Q}\\ \end{array} \right.$$ is discontinuous in every $x\in \mathbb{R}$ but $D(D(x))=1$ is $C^\infty$.
$C^\infty$ means the function is arbitrary often continuous differentiable.