Continuity of Topological product of spaces

Let be an element in the index set I

Let X = _{ }X_ be the topological product of the family of spaces {X_ }. Prove that a functon f:Y->X from a space Y into the product X is continuous if and only if for each I the function f_ = p_ f: Y->X_

I think p_i is defined in a product space as the ith projection p_i: X -> X_i such that p_i(a) = a_i.

Wouldn't this be true because of the definition of continuity? Since a function is uniformly continuous if and only if it is continuous at all points?