Question

Is y=x^(2/3) continuous and differentiable?

Answer 1

It is continuous, but not differentiable due to x being in the denominator in its derivative. The derivative would not exist at x=0.

Answer 2

Thank you very much! I wasn't sure if it was continuous in the first place since there seems to be a cusp and an imaginary part, but the x in the denominator makes sense!

Answer 3

Ah but it would only reach the imaginary numbers if you used the principle root (a root of a number has 2 parts, a negative and positive; principle root is only the positive). However, if you allow both roots, the graph would actually just touch (0,0) then mirror itself along the y-axis!