Before diving into the muck head first, take a moment to go over some of the basic ideas involving continuity -particularly, uniform continuity. We will lightly cover a few concepts of continuity to provide a base for our discussion. We know from basic calculus and our study of integration, differentiation, and continuity that: on an… Read More Pointwise Limits, Uniform Limits, and Continuity

Now, we learn from calculus that the slope of a function can be found by way of its derivative; what is more fascinating is that we can go further -much further. Definition #1: Let be real valued on an interval . The following holds: 1) If whenever and are in with  , then our function is nondecreasing on… Read More Monotonicity and Derivatives

Open any advanced calculus text or work in any field of analysis, you are likely to relish in the glorious ideas of Agustin-Louis Cauchy and not even know it. Cauchy was born in Paris, France in 1789. He began his professional life as an engineer and progressively influenced mathematics by way of research, professorships (when… Read More Cauchy