P ≠ BPP: A Proof via Complexity Function Continuity Analysis

We prove that P ≠ BPP by demonstrating that when BPP problems inherit complexity properties from NP (under the assumption BPP ⊆ NP), they develop discontinuous complexity functions that cannot exist in P. Using the continuity framework established in our proof of P ≠ NP, we show that P ⊆ P_cont (problems with continuous complexity functions), while BPP can contain problems with discontinuous complexity functions under realistic complexity-theoretic assumptions. Since continuous functions cannot equal discontinuous functions, this establishes P ≠ BPP and resolves whether randomness provides fundamental computational advantages in polynomial time.