Answer
The transition is smooth.
Work Step by Step
\[
\frac{\left|y^{\prime \prime}\right|}{\left[y^{\prime 2}+1\right]^{3 / 2}}=\kappa(x)
\]
For $0=y,$ we get $0=\kappa$
Along $x^{2}=y,$ since $y^{\prime}=2 x$ and $y^{\prime \prime}=2,$ then
\[
\begin{aligned}
&\frac{\left|y^{\prime \prime}\right|}{\left(y^{2}+1\right)^{3 / 2}}= \kappa \\
&=\frac{2}{\left[4 x^{2}+1\right]^{3 / 2}}
\end{aligned}
\]
So $2=\kappa(0),$ and then $\kappa$ isn't continuous; hence, the transition isn't smooth
Along $x^{3}=y,$ since $y^{\prime \prime}=6 x,$ and $3 x^{2}=y^{\prime}$:
\[
\begin{aligned}
&\frac{\left|y^{\prime \prime}\right|}{\left(y^{2}+1\right)^{3 / 2}}=\kappa \\
&=\frac{6 x}{\left[9 x^{4}+1\right]^{3 / 2}}
\end{aligned}
\]
$\kappa(0)=0,$ so $\kappa$ is continuous; hence, the transition is smooth.
