If $0<\mathrm{a}<1$ and $\text{⨍ }$ is defined as

$\text{⨍ (x)=}\left\{\begin{array}{ccc}{\mathrm{x}}^{\mathrm{a}}& \mathrm{if}& \mathrm{x}>0\\ 0& \mathrm{if}& \mathrm{x}\le 0\end{array}\right.$ then

 

$$\begin{gathered} 0<\mathrm{a}<1 \\ \mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}{\mathrm{x}}^{\mathrm{a}} \mathrm{if} \mathrm{x}>0\\ 0 \mathrm{if} \mathrm{x}\le 0\end{array}\right. \end{gathered}$$

$$\begin{gathered} {\mathrm{f}}^1\left(0^{-}\right)=0, {\mathrm{f}}^1\left(0^{+}\right)=\underset{\mathrm{x}\to 0^{+}}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(0\right)}{\mathrm{x}-0} \\ \therefore {\mathrm{f}}^1\left(0^{+}\right)=\underset{\mathrm{x}\to 0^{+}}{\lim }\frac{{\mathrm{x}}^{\mathrm{a}}-0}{\mathrm{x}-0} \\ =\underset{\mathrm{x}\to 0^{+}}{\lim }\frac{1}{{\mathrm{x}}^{1-\mathrm{a}}}=\infty \\ \therefore \mathrm{f} \mathrm{is} \mathrm{not} \mathrm{diffrentiable} \mathrm{at} \mathrm{x}=0 \end{gathered}$$