Fixed Points of ODE: Clarifying Conditions

[Apteronotus]

In a book on synchronization it is stated that given the ODE

$$\frac{d\psi}{dt}=-\nu+\epsilon q(\psi)$$

there is at least one pair of fixed points if

$$\epsilon q_{min}<\nu<\epsilon q_{max}$$

were $$q_{min}, q_{max}$$ are the min and max values of $$q(\psi)$$ respectively.

While this could be true under particular circumstances (ie. when $$q_{min}<0, q_{max}>0$$), I don't see how it could hold in general; such as the case when $$q(\psi)>0$$.

Can anyone shed some light on this?

Thanks in advance.

 

[trambolin]

Just think about this...

$$0 = -\nu+\epsilon q(\psi)$$

 

[HallsofIvy]

Assuming that q is continuous, the "intermediate value property" gives the answer.

 

[Apteronotus]

Yes, of course! Thank you both.