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The main result of the paper characterizes continuous local derivations on a class of commutative Banach algebra $A$ that all of its squares are positive and satisfying the following property: Every continuous bilinear map $\Phi$ from $A\times A$ into an arbitrary Banach space $B$ such that $\Phi (a,b)=0$ whenever $ab=0$, satisfies the condition $\Phi (ab,c)=\Phi (a,bc)$ for all $a,b,c\in A$.