EDB — 161

[161]Prerequisites:[0PT]↺↻. Let \((X_ 1,d_ 1)\), \((X_ 2,d_ 2)\) and \((Y,𝛿)\) be three metric spaces; consider the product \(X=X_ 1× X_ 2\) equipped with the distance \(d(x,y)=d_ 1(x_ 1,y_ 1)+d_ 2(x_ 2,y_ 2)\).   ¹ Let \(f:X→ Y \) be a function with the following properties:

• For each fixed \(x_ 1∈ X_ 1\) the function \(x_ 2↦ f(x_ 1,x_ 2)\) is continuous (as a function from \(X_ 2\) to \(Y\));

• There is a continuity modulus \(𝜔\) such that

  \[ ∀ x_ 1,\tilde x_ 1∈ X_ 2~ ~ ,~ ~ ∀ x_ 2∈ X_ 2~ ~ , 𝛿\big(f(x_ 1,x_ 2),f(\tilde x_ 1,x_ 2)\big) ≤ 𝜔\big(d_ 1(x_ 1,\tilde x_ 1)\big) \]

  (We could define this property by saying that the function \(x_ 1↦ f(x_ 1,x_ 2)\) is uniformly continuous, with constants independent of the choice of \(x_ 2\)).

Then show that \(f\) is continuous.