I am trying to prove that $\displaystyle \sqrt[3]{id_\mathbb{R}}$ function is continous.
Any help would be appreciated.
What precisely are you having trouble with? Notice that $\displaystyle x-y=\left(\sqrt[3]{x}\right)^3-\left(\sqrt[3]{y}\right)^3=\left(\sqrt[3]{x}-\sqrt[3]{y}\right)\left(x^{\frac{2}{3}}+\sqrt[3]{xy}+y^{\frac{2}{3}\right)$