Is it decidable that any two computable function over reals $ f(x_1,x_2,\dots,x_n)\equiv g(x_1,x_2,\dots,x_n)$

Question

Is it decidable that any two computable function over reals or over sphere of complex $ f(x_1,x_2,\dots,x_n)\equiv g(x_1,x_2,\dots,x_n)$ ?

Answer 1

No. We can, in classic computer science style, reduce this to the halting problem. In particular, choose some program $P$ and define $f(t)$ to be $1$ if $P$ halts by step $t$ and to be $0$ if it does not. Define $g(t)$ to be $1$ if $P$ halts by step $2t$ and to be $0$ is it does not. It is obvious that if $f(t)=g(t)$ for all $t$, then we have "$P$ halts by step $t$ if and only if it halts by step $2t$" and, easily therefrom, "$P$ does not halt". However, both functions are computable, since we can simple run the program for the requisite number of steps to find out whether it halts by step $t$ or $2t$.