real-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

I am currently reading Axler's Measure, Integration &amp; Real Analysis and I noticed that there does not seem to be a definition for infinite limits, i.e. what it means for a sequence $a_k$ to have ...

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 A Cauchy real number is a real number that is given as…