I like the calculus answers above such as evaluating integrals and infinite series as values of functions satisfying differential equations. The following are much more specialized but seems also to illustrate the point in the question, of deciding something about an individual case by virtue of its membership in a family.
Mumford's proof that a smooth cubic surface has exactly 27 lines is of this type. He proves that cubic surfaces form a family, that the number of lines is finite and constant precisely on the connected family of smooth surfaces. Then he computes the number on one smooth surface.
Hershel Farkas conjectured in 2004 that one can decide whether a 4 dimensional principally polarized abelian variety with an isolated "vanishing theta null" (an isolated double point on its theta divisor at a point of order two) is or is not the Jacobian of a genus 4 curve, by noting whether the rank of the double point is three or four.
One way to check this is as follows:
i) in the universal family of 4 dimensional ppav's, generic Jacobians are characterized by having two ordinary double points on theta.
ii) an isolated double point at a point of order two splits into two odp's under deformation in any reasonably general family if and only if its rank is three.
Thus a ppav with the stated hypotheses is a limit of Jacobians, hence is a Jacobian.
The infinitesimal approach to the Torelli problem is another example. Classically one tried to show the Torelli map t:M(g)-->A(g) (assigning to a curve its Jacobian) is injective by using information in the image point t(C) = J(C) to reconstruct C, either the theta divisor or a birational model. This did not take advantage of the fact that C belongs to a family M(g) and that t is defined on the whole family.
The infinitesimal approach uses the fact that the Torelli map is defined also at all curves near C, and thus uses information in the image of the differential, or in the kernel of the codifferential t*, to reconstruct C.
This makes the problem of finding equations for C much easier since in general the kernel of t* contains all quadrics containing the canonical model of C, while from J(C) one can only reconstruct such quadrics of rank ≤ 4.
In some sense all proofs by induction have this form. I.e. one proves a non trivial statement by checking it in a trivial case n=1, then bootstrapping up one case at a time by induction. Indeed the algebraic geometry example above of finding a curve of every genus with no automorphisms is a geometric form of induction from the easy case of genus 2. The inductive step is the upper semicontinuity statement.