Why does continuity still feel weird?

[RicoGerogi]

Been working through some advanced calc problems and the definition of continuity keeps feeling kinda weird. The epsilon-delta thing doesn't quite click intuitively for all cases, especially with those piecewise functions. Seems like there's a persistent disconnect between the formal proof and practical visualization...

 

[PeroK]

Been working through some advanced calc problems and the definition of continuity keeps feeling kinda weird. The epsilon-delta thing doesn't quite click intuitively for all cases, especially with those piecewise functions. Seems like there's a persistent disconnect between the formal proof and practical visualization...

An alternative, which is equivalent to the usual epsilon-delta definition is:

A function $f$ is continuous at a point $a$ if for every sequence (in the domain of $f$) that converges to $a$, the sequence of function values converges to $f(a)$.

This is often more useful practically for proving that a function is not continuous at $a$, as all you have to do is find a sequence $x_n \to a$ where $f(x_n)$ does not converge to $f(a)$.

That said, you need to know the epsilon definition of convergence for a sequence.