Proving Topology Continuity for F: X x Y -> Z in Separate Variables

[tomboi03]

Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y₀ in Y, the map h: X-> Z defined by h(X)= F( x x y₀) is continuous, and for each x₀ in X, the map k: Y-> Z defined by k(y) =F(x₀ x y) is continuous. Show that if F is continuous, then F is continuous in each variable separately.

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[CompuChip]

What is the definition of F being continuous?