An in-depth analysis of nonautonomous bifurcations of saddle-node
type for scalar differential equations $x'=-x^2+q(t)\,x+p(t)$,
where $q\colon\R\to\R$ and $p\colon\R\to\R$ are bounded and uniformly
continuous, is fundamental to explain the absence or occurrence of
rate-induced tipping for the differential equation
$y' =(y-(2/\pi)\arctan(ct))^2+p(t)$ as the rate $c$ varies on $[0,\infty)$.
A classical attractor-repeller pair, whose existence for $c=0$ is assumed,
may persist for any $c>0$, or disappear for a certain critical rate $c=c_0$,
giving rise to rate-induced tipping. A suitable example demonstrates that
this tipping phenomenon may be reversible.
