What does this theorem of continuity of elementary function means?

Question

I saw in my lecture not about this theorem of elementary function continuity:” All elementary functions are C1 in the interior of their maximal domain except the norm when the argument is zero and the powers with exponent $\alpha \in (0,1)$ when the argument is 0 and 0 is not at the boundary of the maximal domain” I really don’t know what does it means saying “ except the norm when the argument is zero and the powers with exponent $\alpha \in (0,1)$ when the argument is 0 and 0 is not at the boundary of the maximal domain” what is it argument of the norm? And can anyone give me specific examples on these exceptions? Thank you so much!

Answer 1

For example $ |x|$ is not differentiable at 0, nor are $x^p, p\in (0,1)$