The paper studies relative computability of continuous real functions w.r.t. (finite) sets, \(A\), of number-theoretic total functions. The approach generalizes definitions of computability by M. B. Pour-El and J. I. Richards [Computability in analysis and physics. Perspectives in Mathematical Logic, Berlin etc.: Springer-Verlag (1989; Zbl 0678.03027)]. The main result shows that every \(C^1\) function on a compact real interval which is computable in \(A\) has a derivative computable in \(A'\), the jump of \(A\).