A function $f$ such that the limit of $f(x^2)$ exists but not $f(x)$.

Question

I want to show a function $f$ such that $\displaystyle\lim_{x\to x_0}f(x^2)\in\mathbb{R}$ but $\displaystyle\lim_{x\to x_0}f(x)$ doesn't exist.

I only need a suggest of such a function $f$. I can't find an example of this.

Any hint? Thanks.

Answer 1

Let $f(x) = \begin{cases} -1,&x < 0,\\1,& x \geq 0, \end{cases}$ with $x_0 = 0$.