Q4Mathematics I

Cauchy's rigorous $(ϵ-\delta )$ definition forms the absolute foundational bedrock of real analysis and modern calculus. It formally defines continuity without relying on intuitive graphical notions of "unbroken curves."

In strict mathematical terms, this means we can constrain the output $f(x)$ to be arbitrarily close to $f(c)$ (within a tolerance of $ϵ$) simply by forcing the input $x$ to be sufficiently close to $c$ (within a boundary of $\delta$).