I'm currently having trouble to verify this fact on page 90 in Robert L. Devaneys "An Introduction to Chaotical Dynamical Systems":
Let $F(x,\lambda) = f_\lambda (x)$. Assume $f_\lambda (x) = 0$ for all $\lambda$ in an Interval around $\lambda_0$ and $f'_{\lambda_0}(0) = -1$. Consider the function $G(x, \lambda) = f_\lambda(f_\lambda(x)) - x$.
We set \begin{equation*} H(x, \lambda) = \begin{cases} \frac{G(x,\lambda)}{x} & x \neq 0 \\ \frac{\partial G}{\partial x}(0,\lambda) & x = 0 \end{cases} \end{equation*}
where $G(x,\lambda)$ is smooth in both variables. It is now said that one easily verifies that H is smooth and satisfies $$\frac{\partial H}{\partial x} (0,\lambda_0) = \frac{1}{2} \frac{\partial^2 G}{\partial x^2} (0,\lambda_0)$$ and $$\frac{\partial^2 H}{\partial x^2} (0,\lambda_0) = \frac{1}{3} \frac{\partial^3 G}{\partial x^3} (0,\lambda_0)$$ It is clear to me that you only have to check for smoothness when $x=0$, since $G$ is assumed to be smooth. I tried to formally calculate the partial derivatives for $x=0$ using the differential quotient, but I can't seem to get to a conclusion, neither for smoothness, nor for the derivatives in $(0,\lambda_0)$. Am I missing something here or am I just bad at calculating?
Any help would be appreciated!
Note that since $G$ is smooth, $\partial _\lambda $ and $\partial_x$ commute for $G$. Now, in a neighbourhood around $x =0$ but not at $0$, $$H(x,\lambda) = \frac{1}{x}\sum_{k=0}^\infty \frac{x^k}{k!} \frac{\partial^k}{\partial x^k}G(0,\lambda) = \frac{\partial}{\partial x}G(0,\lambda) + \frac{x}{2!} \frac{\partial^2}{\partial x^2}G(0,\lambda) + \frac{x^2}{3!} \frac{\partial^3}{\partial x^3}G(0,\lambda) + \dots$$
where the equality requires $G(0,\lambda) = 0$. This implies that in an interval around $x =0$ but not at $0$,
$$\frac{\partial^m}{\partial x^m} H(x,\lambda) = \frac{1}{m} \frac{\partial^{m+1}}{\partial x^{m+1}} G(0,\lambda) + \frac{x}{(m+1)}\frac{\partial^{m+2}}{\partial x^{m+2}} G(0,\lambda) + \frac{x^2}{2!(m+2)}\frac{\partial^{m+3}}{\partial x^{m+3}} G(0,\lambda) + \dots$$ Now the task reduces to showing that these are continuous at $x = 0$, since the derivatives w.r.t lambda are well defined and smooth already, and further, since the derivatives commute for $G$, the smoothness of $x$-derivatives will imply the smoothness of all derivatives (you should flesh this out a little).
Okay, so, $$\frac{\partial}{\partial x} H(0,\lambda) = \lim_{\epsilon \to 0} \frac{ H(x, \lambda) - \partial_xG(0,\lambda)}{\epsilon} = \lim_{\epsilon \to 0} \frac{ \sum_1^\infty \epsilon^{k}/(k+1)!G(0,\lambda) \partial_x^{k+1}}{\epsilon} = \frac{1}{2} \frac{\partial^2}{\partial x^2} G(0,\lambda)$$
so we have the continuity of the first derivative. Now proceed by induction - assume the $m^{th}$ derivative is continuous at $0$, and show that this implies that the $m+1^{th}$ derivative exists and is continuous at $0$. I'll leave this bit to you.