Let mathcal{A} be an algebra and delta, varepsilon: mathcal{A} to mathcal{A} be linear mappings. We say that a linear mapping d: mathcal{A} to mathcal{A} is a ( delta, varepsilon) -double derivation if d(ab)=d(a)b+ad(b)+ delta(a) varepsilon(b)+ varepsilon(a) delta(b) for all a,b in mathcal{A} . By a delta -double derivation we mean a delta, delta) -double derivation. Giving some elementary facts concerning double derivations, we prove that if mathcal{A} is a C^* -algebra, delta: mathcal{A} rightarrow mathcal{A} is a -linear mapping and d: mathcal{A} rightarrow mathcal{A} is a continuousdelta -double derivation then delta is continuous. We also show that if mathcal{A} is a C^* -algebra, delta: mathcal{A} rightarrow mathcal{A} is a continuous linear mapping and d: mathcal{A} rightarrow mathcal{A} is a * - delta -double derivation then d is continuous. Similar facts concerning ( delta,varepsilon) -double derivations on C^* -algebras are also given.

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