why here $x^2$ is used ? why not $x ?$ 
Does  there exists a function $f : \mathbb{R }\rightarrow \mathbb{R}$ which is differentiable only at the point $0.$?

My attempt :  I found the  answer  here Is there a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable?
But  i didn't understands the  answer , my doubts given below


The function with $x$ is not differentiable at $0$. For instance in the limit defining the derivative, the $\lim \sup$ exists and is $1$, and the $\lim \inf$ exists and is $0.$ These are not equal, so the derivative doesn't exist at $0.$
