Question

Is the function f : R ➡R, defined by f(x) = 1-|x| is differentiable in its domain.

Answer 1

Let us check whether the function $f : \R \to\R$ , defined by $f(x) = 1-|x|$ is differentiable in its domain.

Since $\lim\limits_{x\to 0+}\frac{f(x)-f(0)}{x-0} =\lim\limits_{x\to 0+}\frac{1-|x|-1}{x} =\lim\limits_{x\to 0+}\frac{-|x|}{x} =\lim\limits_{x\to 0+}\frac{-x}{x}=-1$ and

$\lim\limits_{x\to 0-}\frac{f(x)-f(0)}{x-0} =\lim\limits_{x\to 0-}\frac{1-|x|-1}{x} =\lim\limits_{x\to 0-}\frac{-|x|}{x} =\lim\limits_{x\to 0-}\frac{-(-x)}{x} =\lim\limits_{x\to 0-}\frac{x}{x}=1\ne -1,$

we conclude that the function $f$ is not differentiable at the point $x=0$ , and hence it is not differentiable in its domain.