Let's say I've got a function $f$ that takes a single number and returns a number. And I have another function $\mathrm{verify}f$ which takes the input I gave to $f$ and the number returned by $f$ which returns true if the output is the same as $f$ would have returned given the input.If you know the input and output to $f$ and the function to verify then it is possible to work out what the implementation of $f$ is.What I'd like to know, is if there exist some function that even if you know the output and the input and the verification function it's impossible to work out what $f$ does.It feels like these functions must not exist – but I don't know that for sure.
Function whose implementation is difficult (computationaly) to work out
computability