Discuss the applicability of Lagrange's mean value theorem for the function

Given:

$f(x)=|x|$

If Lagrange's theorem is applicable for the given function, then $f(x)$ is continuous on $[-1,1]$ and differentiable on $(-1,1)$.

But it is known that $f(x)=|x|$ is not differentiable at $x=0 \in(-1,1)$.

Thus, our supposition is wrong.
Therefore, Lagrange's theorem is not applicable for the given function.