[15] Let (t, x) be a computable function and limt (t, x) = C(x), for all x. For each t define t(x) = (t, x)

[15] Let φ(t, x) be a computable function and limt→∞ φ(t, x) =

C(x), for all x. For each t define ψt(x) = φ(t, x) for all x. Then C is the limit of the sequence of functions ψ1, ψ2,... . Show that for each error

 and all t there are infinitely many x such that ψt(x) − C(x) > .

Comments. C(x) is the uniform limit of the approximation above if for each  > 0, there exists a t such that for all x, ψt(x) − C(x) ≤ . The exercise shows that C(x) is not the uniform limit of the above sequence of functions.