  Critical transitions are observed in many complex systems. This includes the
onset of synchronization in a network of coupled oscillators or the emergence
an epidemic state within a population. "Explosive" first-order transitions have
caught particular attention in a variety of systems when classical models are
generalized by incorporating additional effects. Here we give a mathematical
argument that the emergence of such first-order transitions is not surprising
but rather a universally expected effect: Varying a classical model along a
generic two-parameter family must lead to a change of the criticality. To
illustrate our framework, we give three explicit examples of the effect in
distinct physical systems: a model of adaptive epidemic dynamics, for a
generalization of the Kuramoto model, and for a percolation transition.
