continuity of composition of functions

All functions in this entry are functions from $\\mathbb{R}$ to $\\mathbb{R}$ .

Example 1Let $f(x)=1$ for $x\\leq 0$ and $f(x)=0$ for $x>0$ ,let $h(x)=0$ when $x\\in\\mathbb{C}$ and $1$ when $x$ is irrational,and let $g(x)=h(f(x))$ . Then $g(x)=0$ for all $x\\in\\mathbb{R}$ , sothe composition of two discontinuous functions can becontinuous.

Example 2If $g(x)=h(f(x))$ 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 3Suppose $g(x)=h(f(x))$ is continuous and $f$ is continuous.Then $h$ does not need to be continuous. For a conterexample, put $h(x)=0$ for all $x\eq 0$ , and $h(0)=1$ , and $f(x)=1+|x|$ . Now $h(f(x))=0$ is continuous, but $h$ is not.

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

单词	ContinuityOfCompositionOfFunctions
释义	continuity of composition of functions All functions in this entry are functions from $\\mathbb{R}$ to $\\mathbb{R}$ . Example 1Let $f(x)=1$ for $x\\leq 0$ and $f(x)=0$ for $x>0$ ,let $h(x)=0$ when $x\\in\\mathbb{C}$ and $1$ when $x$ is irrational,and let $g(x)=h(f(x))$ . Then $g(x)=0$ for all $x\\in\\mathbb{R}$ , sothe composition of two discontinuous functions can becontinuous. Example 2If $g(x)=h(f(x))$ 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 3Suppose $g(x)=h(f(x))$ is continuous and $f$ is continuous.Then $h$ does not need to be continuous. For a conterexample, put $h(x)=0$ for all $x\eq 0$ , and $h(0)=1$ , and $f(x)=1+\|x\|$ . Now $h(f(x))=0$ is continuous, but $h$ is not. Example 4Suppose $g(x)=h(f(x))$ is continuous and $h$ is continuous. Then $f$ does notneed to be continuous. For a counterexample, put $f(x)=0$ for all $x\eq 0$ , and $f(0)=1$ , and $h(x)=0$ for all $x$ . Now $h(f(x))=0$ is continuous, but $f$ is not.
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