I have trouble understanding the difference in the following problem. I have the function F(x) which is not differentiable at x=a $$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for $x<a$}\\ G(x) & \text{ for $x\geq a$}\end{array}\right.$$ and i want the derivative of F(x), F'(x) to be continuous in x=a.
I can use a cubic smoothstep function $S_3(x)$ such that $$F_{smooth}(x)=S_3(\frac{x-a}{\epsilon})G(x)$$ which by definition is $$S_3(a)=0, \qquad S_3(a+\epsilon)=1\\ S_3'(a)=0, \qquad S_3'(a+\epsilon)=0$$ such that the derivative of the function is $$F'_{smooth}(x)=S'_{3}G(x)+S_{3}G'(x)$$ So for $F'_{smooth}(a)=0$ and $F'_{smooth}(a+\epsilon)=G'(a+\epsilon)$ which makes the derivative continuous.
Another definition i found is to use cubic hermite splines such that $$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for x<a}\\ P_{3}(x) & \text{ for $a\leq x\leq a+\epsilon$}\\ G(x) & \text{ for $x\geq a+\epsilon$}\end{array}\right.$$
where the cubic polynom $P_{3}(x)$ is defined with $$P_{3}(a)=0 \quad P_{3}(a+\epsilon)=G(a+\epsilon)\\ P'_{3}(a)=0 \quad P_{3}'(a+\epsilon)=G'(a+\epsilon)$$ I plotted both smoothing functions and they look differently.\ So what is the actual difference between these two approaches? Or are they equally? From their definitions, they lead to the exact same values for $x<a$ and $x>a+\epsilon$ but within $[a,a+\epsilon]$ the functions are $$ \begin{array}{cl}F1_{smooth}(x)=S_{3}G(x) & F1'_{smooth}(x)=S'_{3}G(x)+S_{3}G'(x)\\ F2_{smooth}(x)=P_{3}(x) & F2'_{smooth}(x)=P'_{3}(x)\end{array}$$
If they aren't the same, what are the advantages and disadvantages?