Is there a connection between condition number of a problem (a problem is a function from one normed vector space X of data to another normed vector space Y of solutions ) and radius of curvature of the function? In other words, can we infer whether a problem is ill- or well conditioned near a point x in X just by calculating the radius of curvature of the function near that point x?

For example, lets consider f(x) = tan (x). We want to determine whether this problem is ill- or well conditioned near x=pi/2. If we calculate the condition number with the help of the Jacobian, we can readily infer that the condition number is very very large which makes the problem ill-conditioned. Now let's take on a different angle of reasoning.

In the small left neighborhood of x=pi/2, a small change in x will result in a large change of f(x), which makes the problem ill-conditioned near pi/2. This massive change in f(x) near pi/2 results from the radius of curvature of the function near pi/2. Thus from the order of magnitude of the radius of curvature gives the information of the conditioning of any problem.

Any suggestions or feedback would be highly appreciated.