[joe_the_user]

The function has a derivative by some notions of derivative.

But the function's derivative can't be derived by an application of the chain rule and the know derivatives of primitive functions, which is what Algorithmic/automatic differentiation ultimately does (though it does this at run time, not compile time, since ordinary, symbolic differentiation explodes in memory for a complicated functions).

Also:

The continuous function

    int f(int x) {
        if(x > 2) 
            return x + x;
        else 
            return x*2;
    }

Is not automatically or symbolically differentiable when represented that way.

[Certhas]

I believe a source to source differentiator could deal with all these (where well defined of course), e.g.:

https://github.com/FluxML/Zygote.jl

    julia> fs = Dict("sin" => sin, "cos" => cos, "tan" => tan);
    
    julia> gradient(x -> fs[readline()](x), 1)
    sin
    0.5403023058681398