It has been a year or so I took my course of real analysis, still could not understand these two definitions of continuity-(These two definitions are given as chapter 9 and 10 in the classroom resource material of MAA -Exploratory Examples in Real Analysis).
Definition 1:Sequence definition of continuity
$f(x_0)$ exists;
$\lim_{x \to x_o} f(x)$ exists; and
$\lim_{x \to x_o} f(x)$ =$f(x_o)$.
The sequence definition is convinient tool to prove continuity of polynomials. This definition is also useful when in proving discontinuity.
Definition 2:
Let $y=f(x)$ be a function.Let $x=x_o$ be a point of domain of $f$ .The function $f$ is said to be continuous at $x=x_o$ iff given $\epsilon \gt 0$,there exists $\delta \gt 0$ such that if $x \in (x_0-\delta,x_0+\delta)$, then $f(x)\in (f(x_o)-\epsilon,f(x_o)+\epsilon ) $.
This definition is extremely useful when considering a stronger form of continuity,the Uniform Continuity.
I know that Definition 2 puts the use of $\epsilon - \delta$ ,My doubt is the use of a sequence in definition 1 ,i.e. in 1.) Given a sequence $(x_n)_{n=1}^{\infty}$ that converges to domain point $x=x_o$ of $f$ ,we are interested in determining the relationship between the continuity of $f$ at $x=x_o$ and the behaviour of corresponding sequence of outputs $(f(x_n))_{n=1}^{\infty}$ which forms the basis of the definition. How is it equivalent to 2.) Is there some proof for this..
Any help would be appreciated.