All functions in this entry are functions from $\sR$ to $\sR$ .

Example 1 Let $f(x)= 1$ for $x \leq 0$ and $f(x) = 0$ for $x >0$ , let $h(x) = 0$ when $x\in \sQ$ and $1$ when $x$ is irrational, and let $g(x)=h(f(x))$ . Then $g(x)=0$ for all $x\in \sR$ , so the composition of two discontinuous functions can be continuous.

Example 2 If $g(x) = h(f(x))$ is continuous for all functions $f$ , then $h$ is continuous. Simply put $f(x)=x$ . Same thing for $h$ and $f$ . If $g(x) = h(f(x))$ is continuous for all functions $h$ , then $f$ is continuous. Simply put $h(x)=x$ .

Example 3 Suppose $g(x) = h(f(x))$ is continuous and $f$ is continuous. Then $h$ does not need to be continuous. For a conterexample, put $h(x)=0$ for all $x\neq 0$ , and $h(0)=1$ , and $f(x)=1+|x|$ . Now $h(f(x))=0$ is continuous, but $h$ is not.

Example 4 Suppose $g(x) = h(f(x))$ is continuous and $h$ is continuous. Then $f$ does not need to be continuous. For a counterexample, put $ f(x) = 0$ for all $x\neq 0$ , and $f(0)=1$ , and $h(x)=0$ for all $x$ . Now $h(f(x))=0$ is continuous, but $f$ is not.