continuity of composition of functions

All functions in this entry are functions from $ℝ$ to $ℝ$.

Example 1 Let $f(x)=1$ for $x\le 0$ and $f(x)=0$ for $x>0$, let $h(x)=0$ when $x\in ℂ$ and $1$ when $x$ is irrational, and let $g(x)=h(f(x))$. Then $g(x)=0$ for all $x\in ℝ$, 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 $x\ne 0$, 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 $x\ne 0$, and $f(0)=1$, and $h(x)=0$ for all $x$. Now $h(f(x))=0$ is continuous, but $f$ is not.

Title	continuity of composition of functions
Canonical name	ContinuityOfCompositionOfFunctions
Date of creation	2013-03-22 14:04:55
Last modified on	2013-03-22 14:04:55
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	7
Author	bbukh (348)
Entry type	Result
Classification	msc 54C05
Classification	msc 26A15