I'm working on some questions on continuity, and one question was to define what it meant for a function $\ f:[a, b]→ℝ$ to be continuous on $\ [a,b]$.
The definition given is : $\ f:[a, b]→ℝ$ is continuous on $\ [a, b]$ if, $\ \forall\ c\ \epsilon\ [a,b]$ and $\ \forall\ (x_{n})_{n=1}^{\infty} \subset [a,b]$ such that $\ \lim_{n→\infty} (x_{n}) = c $, it follows that $\ \lim_{n→\infty}f(x_n) = f(c)$.
This makes sense to me, and I'm more than happy with the definition. However, when I answered this question I wrote :
$\ f$ is continuous on$\ [a,b]$ if $\ \forall\ c\ \epsilon\ [a,b]$, given $\ ε > 0, \exists\ \delta > 0$ such that : $\ |x-c| < \delta \implies |f(x) - f(c)| < ε$.
Are these definitions equivalent? Is there a problem with my definition? I see that the difference arises from me using the epsilon-delta definition of coninuity, but does this break anything?
Thanks.