A closed unbounded ∗-derivation on a C^{∗}-algebra A.

******************************

Suppose that A = C([0, 1]) and δ(f) = (\frac{d}{dt})f(t) = f′(t) with the domain D(δ) = C^{1}([0, 1]) where C^{1}([0, 1]) is the algebra of all continuously differentiable functions on [0, 1]. ∥ δ(x^{n}) ∥= n = n ∥ x^{n} ∥ implies that δ is an unbounded derivation from D(δ) into A. If f_{n} ∈ D(δ), f_{n} ⟶ f ∈ A, and δ(f_{n}) ⟶ g, then f′_{n} ⟶ g uniformly on [0, 1] and f_{n}(0) ⟶ f(0). So g is differentiable and f′ = g. Therefore f ∈ D(δ) and δ(f) = g.