Continuity of Topological product of spaces

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

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?

Use the fact that the projection is open.

Quote:

Since a function is uniformly continuous if and only if it is continuous at all points?

This is not true. The exponential function is continuous at every real, but it is not uniformly continuous over the reals.