Continuous Sensitivity and Reversibility

Let \(n\) be a positive integer and \(f\) a differentiable function from a convex subset \(C\) of the Euclidean space \(\mathbb{R}^n\) to a smooth manifold. We define an invariant of \(f\) via counting certain threshold functions associated to \(f\). We call this invariant the continuous sensitivity of \(f\) and denote it by \(\mathrm{cs}_{C}(f)\). This invariant is a real number between \(0\) and \(n\) and measures how sensitive \(f\) is to change in its input variables. For example, if \(f\) is a constant function then \(\mathrm{cs}_{C}(f)=0\). On the other extreme, if \(\mathrm{cs}_{C}(f)=n\) then \(f\) is one-to-one on \(C\). This last statement is important for reversibility problems. To say that a function is reversible one can write an explicit inverse of the function. However, this is not always easy. Even a multilinear function can have a complicated inverse function. Here we give tools to compute continuous sensitivity which makes it possible to answer reversibility problems without finding explicit inverse functions.