# Continuous but not differentiable

• Sep 1st 2009, 10:16 AM
DaRush19
Continuous but not differentiable
Give an example of 2 continuous functions $\displaystyle f : R \rightarrow R$ satisfying $\displaystyle f(0)=g(0)=0$ and such the composition $\displaystyle f \circ g$ is differentiable at $\displaystyle x=0$ but neither f nor g is differentiable at $\displaystyle x=0$.
• Sep 1st 2009, 10:19 AM
DaRush19
Quote:

Originally Posted by DaRush19
Give an example of 2 continuous functions $\displaystyle f : R \rightarrow R$ satisfying $\displaystyle f(0)=g(0)=0$ and such the composition $\displaystyle f \circ g$ is differentiable at $\displaystyle x=0$ but neither f nor g is differentiable at $\displaystyle x=0$.

This was meant to say: Give an example of 2 continuous functions $\displaystyle f : R \rightarrow R$ and $\displaystyle g: R \rightarrow R$ satisfying...
• Sep 1st 2009, 10:24 AM
Enrique2
Define $\displaystyle f(x)=g(x)=\frac{1}{x}$ if $\displaystyle x\neq 0$ and $\displaystyle f(0)=g(0)=0$. We have $\displaystyle f\circ g(x)=x$
for all $\displaystyle x\in \mathbb{R}$. Neither $\displaystyle f$ nor $\displaystyle g$ is continuous at 0.