Simple function which has domain all real numbers and does not have derivation in point 0.

Question

does $x^{1/3}$ have a derivative at $0$? I mean when we calculate it by definition, limits of both sides is $+\infty$ but I am not sure about conclusion, can anybody please explain it to me? thx a lot

Answer 1

That function is not differentiable at $0$ because asserting that a function $f$ is differentiable at a point $a$ means that the limit $\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ exists in $\mathbb R$. In your case, the limit exists, yes, but it is not a real number.

Answer 2

No, it does not, exactly as you said, the limit is undefined, hence it does not have a derivative