Continuity is defined in my book Basic Analysis (by Lebl pg 86) like this:
Let $S \subset \mathbb R$, $f : S\rightarrow \mathbb R$ be a function, and let $c \in S$ be a number. We say that $f$ is continuous at c if for every $\epsilon > 0$ there is a $\delta > 0$ such that whenever $x \in S$ and $|x − c| < \delta$ , then $| f (x) − f (c)| < \epsilon$.
When $f : S \rightarrow \mathbb R$ is continuous at all $c \in S$, then we simply say that f is a continuous function.
But how does this definition ensure continuity?
If, for $c=0,\epsilon = 10$ there exists $\delta = 5$ for which $|x-0| = |x| < 5, |f(x)-f(c)| < 10$ , and for another $\epsilon=30$, there exists $\delta = 1$ for which $|x| < 1, |f(x)-f(c)| < 30$, wouldnt this increase chances of a function that wildly jumping between values, and perhaps indicating a noncontinous function? I just picked these values out of a hat of course, but the definition does not say anything about how values are structured, if they become smaller while approaching $c$ for example.