Travelling wave ODE: showing function is continuous and decreasing

For $$M>0, c\in\mathbb{R}$$ consider the boundary value problem $$\begin{cases}y''+cy'+f(y)=0,\quad~ -M where $$f$$ is some non-linearity which is Lipschitz and $$f(0)=f(1)=0$$.

For each pair $$(M,c)$$ there is a unique solution $$y(\cdot,M,c)$$ which is monotonous.

(I can assume this fact.)

Show that, for fixed $$M$$, the function $$c\mapsto y$$ solution of \ref{Eq} is continuous and decreasing.

Hint: Consider two solution $$(y_1,c_1)$$ and $$(y_2,c_2)$$ with $$c_2>c_1$$ and plug $$y_1$$ into Eqn(M,$$c_2$$).

Concerning the continuity

Isn't it true that for fixed $$M$$ the solutions depend continuously on $$c$$ (isn't that a general thing)? And doesn't this already imply the continuity of the given function?

Concerning the decrease

If I plug $$y_1$$ into Eqn(M,$$c_2$$), what I get is $$y_1''+c_2y_1'+f(y_1)\neq 0$$

(This cannot be equal to $$0$$, since $$y_1$$ is the unique solution to Eqn(M,$$c_1$$)...)

To be honest, I do not see how to use the given hint.