If the solution of the differential equation $$\dfrac{dy}{dx}=f(x,y,\lambda)$$ under initial conditions $x_{0},y_{0}$, is continuously dependent on the parameter $\lambda$, does it imply that it will also be continuously dependent on the initial conditions $x_{0},y_{0}$.
I came across an argument that essentially said that the equation may be re written as $$\dfrac{d\alpha}{d\beta}=f(x_{0}+\beta,y_{0}+\alpha,\lambda)$$ where $\alpha =y-y_{0}$ and $\beta=x-x_{0}$. Then it was argued that the function on the right hand side is a continuous function in $x_{0},y_{0}$ and hence the solution will be. I am not able to understand how this argument holds.
In order for this argument to make any sense, you'd have to say that the solution depends continuously on parameters "in general", not on a particular parameter in a particular differential equation. Otherwise, you could take any differential equation you want and introduce a parameter that does nothing at all. The solution does not depend on the parameter at all, and therefore is continously dependent on it.