〈 Berger | Dillon 〉

When thinking about how "nice" a function is, I sometimes get tunnel vision and only look at the differentiability and integrability of a function. There are tons of other nice properties out there that tell us a lot about a function.

For example, if a function is monotone, (increasing over an interval or decreasing over an interval) we know that it has at most countably many discontinuities. Also, a theorem from Lebesgue tells us it is differentiable almost everywhere.

If a function is a contraction mapping, we know it has a unique fixed point and can even get a convergence rate for the iteration x_n = f(x_{n-1}). Useful in the study of methods for solving nonlinear equations.

If a function is convex, there's a whole field dedicated to their properties and more.