counterexample
If someone claims ‘all swans are white’, you do not need a theory to refute them — one black swan settles it forever. A counterexample is that black swan: a single concrete case that demolishes a universal claim.
A counterexample is an explicit instance satisfying the hypotheses of a universal statement but violating its conclusion, thereby proving the statement false. Because a claim of the form ‘for all x, P(x)’ is refuted by the existence of even one x with not-P(x), counterexamples are logically decisive — no count of confirming cases can rescue a claim that has one.
In analysis, counterexamples do far more than refute: they sharpen. When a plausible theorem fails, the counterexample reveals exactly which extra hypothesis is needed to rescue it. The pointwise limit of continuous functions need not be continuous — that counterexample is precisely what motivates the stronger notion of uniform convergence. Searching for a counterexample is also the fastest honesty check on any conjecture you are tempted to believe.
Claim: ‘if a function is continuous, it is uniformly continuous.’ Counterexample: f(x) = 1/x on (0, 1) is continuous but not uniformly continuous — near 0 the same change in x produces ever-larger jumps in f. One example kills the claim and shows compactness of the domain was the missing hypothesis.
The counterexample names the missing hypothesis: a compact domain.