analysis

The definition of a continuous function at a point ##x_0## is: $$\forall \epsilon >0 \exists \delta >0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$$ What would happen if we change it to another defintion as: ##\exists \epsilon >0 \forall \delta>0 \forall x(|x-x_0|<\delta... Read more

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…

Just a brief announcement that I have been working with Quanta Books to publish a short book in popular mathematics entitled “Six Math Essentials“, which will cover six of the fundamental concepts in mathematics — numbers, algebra, geometry, probability, analysis, and dynamics — and how they connect with our real-world intuition, the history of math and science, and to modern practice of mathemat…

I was reading a theorem giving conditions for a divergent series to have a convergent subseries and had a sort of flashback. I studied nonlinear PDEs in grad school, which amounted to applied functional analysis. We were constantly proving or using theorems about sequences having convergent subsequences, often subsequences that converged in a very weak […] The post Convergent subsequence first ap…

A bump function is a smooth (i.e. infinitely differentiable) function that is positive on some open interval (a, b) and zero outside that interval. I mentioned bump functions a few weeks ago and discussed how they could be used to prevent clicks in radio transmissions. Today I ran into a twitter thread that gave a […] The post Bump functions first appeared on John D. Cook .

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … The intermediate value theorem (IVT) is a fundamental principle of analysis which allows one to find a desired value by interpolation. It says that a continuous function from an int…

Also known as Rudin’s Infamous Tiny Torture Box. Trigger Warning: Baby Rudin. I. For my undergrad, I majored in Applied Mathematics at a university that was fairly well known for its mathematics department. One of the constants in “Applied Mathematics” is the use of calculus for a variety of applications. “The real world”. In statistics and … Continue reading Book Review: Principles of Mathematic…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism basic constructions: strong axioms further In real ana…