Let $f\left( x \right)$ be a differentiable function $\mathbb{R}\to \mathbb{R}$. Find all the points where the function $\left| f\left( x \right) \right|$ is not differentiable.
Of course, I understand that $\left| f\left( x \right) \right|$ is not differentiable where its graph has a corner because of the modulus, e.g., ${{\delta }_{+}}f\left( a \right)+{{\delta }_{-}}f\left( a \right)=0$. Obviously, it is only possible where $f\left( x \right)$ changes its sign since $f\left( x \right)$ is be a differentiable function.
Is there any way to write this informal idea in a more formal manner?
Find all the points where a function is not differentiable
derivativesreal-analysis
Best Answer
If $f(x) >0$ then $f>0$ in some open interval $(x-r,x+r)$ (by continuity) so $|f|=f$ in this inetrval and so $|f|$ is differentiable at $x$. Similarly, $f(x) <0$ the $|f|$ is differentiable at $x$. But if $f(x)=0$ we cannot say whether $f$ is differentiable or not at $x$: If $f=0$ the $|f|$ is differentiable at all points but if $f(x)=x$ then $|f|$ is not differentiable at $0$. If $f(x)=0$ then $f$ is differentiable at $x$ iff $f(y)=o(y-x)$ as $ y \to x$.