  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.
