I'm confused by my textbook:
Continuity is first defined using its sequential definition:
(1) A function $f$ defined on $A$ is continuous at a point $a ∈ A$ if for each sequence $(x_n)$ in $A$ such that $x_n → a$, we have $f(x_n) → f(a)$.
and then:
(2) $f$ is continuous (on $A$) if $f$ is continuous at each point $a ∈ A$.
But way later is the $ε-δ$ definition of continuity introduced:
(3) Let the function $f$ have domain $A$ and let $c ∈ A$. Then $f$ is continuous at $c$ if for each $ε > 0$, there exists $δ > 0$ such that: $|f(x) − f(c)| < ε$, for all $x ∈ A$ with $|x − c| < δ$.
and the uniform continuity definition:
(4) A function $f$ defined on an interval $I$ is uniformly continuous on $I$ if for each $ε > 0$, there exists $δ > 0$ such that $|f(x) −f(y)| < ε$, for all $x, y ∈ I$ with $|x − y| < δ$.
It is said then that (1) and (3) are in fact equivalent.
But are (2) and (4) equivalent also?
That is:
Is saying "the function $f$ is uniformly continuous on I"(4) the same as "the function is continuous at each point of $I$" (2)?