In mathematics, the **continuous functional calculus** of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,

**Theorem**. Let *x* be a normal element of a C*-algebra *A* with an identity element e; then there is a unique mapping π : *f* → *f*(*x*) defined for *f* a continuous function on the spectrum Sp(*x*) of *x* such that π is a unit-preserving morphism of C*-algebras such that π(1) = e and π(ι) = *x*, where ι denotes the function *z* → *z* on Sp(*x*).

The proof of this fact is almost immediate from the Gelfand representation: it suffices to assume *A* is the C*-algebra of continuous functions on some compact space *X* and define

Uniqueness follows from application of the Stone-Weierstrass theorem.

In particular, this implies that bounded self-adjoint operators on a Hilbert space have a continuous functional calculus.

For the case of self-adjoint operators on a Hilbert space of more interest is the Borel functional calculus.

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