The discrete Cesàro operator C associates to a given complex sequence s = {s n } the sequence Cs ≡ {b n }, where b n = s 0 +•••+s n n+1 , n = 0, 1,. . .. When s is a convergent sequence we show that {C n s} converges under the sup-norm if, and only if, s 0 = lim n→∞ s n. For its adjoint operator C * , we establish that {(C *) n s} converges for any s ∈ 1. The continuous Cesàro operator, Cf (x) ≡ 1 x x 0 f (s)ds, has two versions: the finite range case is defined for f ∈ L ∞ (0, 1) and the infinite range case for f ∈ L ∞ (0, ∞). In the first situation, when f : [0, 1] → C is continuous we prove that {C n f } converges under the sup-norm to the constant function f (0). In the second situation, when f : [0, ∞) → C is a continuous function having a limit at infinity, we prove that {C n f } converges under the sup-norm if, and only if, f (0) = lim x→∞ f (x).

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