A result in complex analysis stating that any entire (holomorphic on the whole complex plane) function that is bounded must be constant.

Principle

Global boundedness plus holomorphicity on C forces all complex derivatives to vanish, so the function cannot vary.

Demonstration

If f is entire and |f(z)| ≤ M for all z, Cauchy estimates give |f^{(n)}(0)| ≤ n! M / R^n for any R; letting R → ∞ yields all derivatives beyond order zero equal zero, so f is constant.

Misapplication

Applying the theorem to functions that are only bounded on a proper subset of C, to meromorphic functions, or to real-differentiable functions; or concluding boundedness on a domain implies constancy without global holomorphicity.

Consequence

Provides a sharp restriction on the growth of entire functions, underpins the Fundamental Theorem of Algebra (by applying to 1/p(z) for a nonconstant polynomial p), and rules out nonconstant bounded entire behaviors.

Reversal

The contrapositive: any nonconstant entire function must be unbounded on C; equivalently, existence of a nonconstant entire function implies it attains arbitrarily large values.

Boundary

Requires holomorphicity on the entire complex plane and boundedness on C. It does not apply to functions with poles, to functions only holomorphic on a proper domain, or to functions bounded only on subsets or along sequences.

Semantic Tension

Tension arises with real-analysis intuition where bounded differentiable functions can be nonconstant; in complex analysis, holomorphicity is much stronger and interacts with global topology.

Synthesis

Liouville's Theorem ties global holomorphic regularity and boundedness: in the full complex plane these two conditions force triviality, making constant functions the only bounded entire maps.