(calculus) The theorem that any real-valued differentiable function that attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. In mathematical terms, if f:ℝ→ℝ is differentiable on (a,b) and f(a)=f(b) then ∃c∈(a,b):f'(c)=0.

(mathematics) The result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

A theorem stating that the interaction picture does not exist in an interacting, relativistic quantum field theory.

(physics) A no-go theorem stating that, during measurement of an entangled quantum state, it is not possible for one observer, making a measurement of a subsystem of the total state, to communicate information to another observer.

(geometry) A result in plane geometry which cannot be derived from Euclid's postulates. It states that, given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).

(geometry) Pythagorean theorem

(mathematics) A fundamental result in Diophantine approximation to algebraic numbers, stating that these numbers cannot have many rational number approximations that are 'very good' (variously defined through history).

(mathematics) One of the earlier theorems leading to Roth's theorem (a Diophantine approximation to algebraic numbers), establishing an exponent d/2+1+ε.

(computing theory) A theorem stating that it is impossible for a distributed data store to simultaneously provide more than two of three guarantees: consistency (every read receives the most recent write or an error), availability (every request receives a non-error response, without the guarantee that it contains the most recent write), and partition tolerance (the system continues to operate despite an arbitrary number of messages being dropped or delayed between nodes).

(logic) The following theorem of Boolean algebra: XY+X'Z+YZ=XY+X'Z where YZ, the algebraically redundant term, is called the "consensus term", or its dual form (X+Y)(X'+Z)(Y+Z)=(X+Y)(X'+Z), in which case Y+Z is the consensus term. (Note: X+Y,X'+Z⊢Y+Z is an example of the resolution inference rule (replacing the + with ∨ and the prime with prefix ¬ might make this more evident).)