Thomas' Calculus 13th Edition

Step 1. To determine if the piecewise defined function is differentiable at the origin, we need to make sure that the function is continuous at the origin and the left and right derivatives are equal. Step 2. It appears that the function is continuous at the origin and $g(0)=0$ Step 3. Check derivative: right side: $g'(0^+)=\lim_{h\to0^+}\frac{f(0+h)-f(0)}{h}=\lim_{h\to0^+}\frac{h^{2/3}}{h}=\lim_{h\to0^+}\frac{1}{h^{1/3}}=\infty$. There is no need to check further the left-side derivative. Step 4. Since one side of the derivative is $\infty$, the function is not differentiable at the origin.