The epsilon–delta definition of continuity is:
"A function $f(x)$ from $\mathbf{R}$ to $\mathbf{R}$ is continuous at point $x_0 \in \mathbf{R}$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that whenever $|x–x_0| < \delta$ then $|f(x)–f(x_0)| < \epsilon$"
$\text{ }$
The intuitive informal statement of continuity is:
"A function $f(x)$ from $\mathbf{R}$ to $\mathbf{R}$ is continuous at point $x_0 \in \mathbf{R}$ if there is no sudden jump in $f(x)$ in the immediate neighbourhood of point $x_0$"
After wondering for quite some time how on earth does the definition statement imply the intuitive statement, I came up with the following interpretation:
Let us first take $\epsilon=\epsilon_1$.
Therefore there exists a $\delta_1 > 0$ such that whenever $|x–x_0| < \delta_1$ then $|f(x)–f(x_0)| < \epsilon_1$
Therefore $|f(\text{immediate neighbourhood of } x)–f(x_0)| < \epsilon_1$
This guarantees that the discontinuity at $x_0$ is less than $\epsilon_1$
Since $\epsilon_1$ could be "any small number greater than zero", the discontinuity at $x_0$ is less than "every small number greater than zero". That is, the discontinuity at $x_0$ is zero. That is, there is continuity at $x_0$.
I think you meant neighbourhood of $x_0$. But yes, your understanding of the definition seems right, especially at the end when you talk about the discontinuity at $x_0$ being less than $\epsilon_1$, and then concluding the discontinuity at $x_0$ has to be $0$.
If you're interested, you should read up about the oscillation of a bounded function at a point. This is defined as follows: let $A \subset \Bbb{R}$ be non-empty, (or more generally, a non-empty subset of $\mathbb{R}^n$), fix a point $a \in A$, and let $f: A \to \Bbb{R}$ be a bounded function. For $\delta > 0$, define the following quantities: \begin{align} M(a,f,\delta) := \sup \{f(x)|\, x \in A \text{ and } |x-a|< \delta \} \\ m(a,f,\delta) := \inf \{f(x)|\, x \in A \text{ and } |x-a|< \delta \} \end{align}
Since we assumed $f$ is bounded, these quantities exist. Now, define the oscillation of $f$ at $a$, denoted $o(f,a)$ by \begin{equation} o(f,a) := \lim_{\delta \to 0} \left[M(a,f,\delta) - m(a,f,\delta) \right] \end{equation} Then, one can prove that $f$ is continuous at $a$ if and only if $o(f,a) = 0$.
Thus, the oscillation of a function at a point is a precise way of seeing how much a bounded function fails to be continuous; in a sense, this is the precise formulation of your last paragraph.
For more information about oscillation, see Spivak's Calculus on Manifolds, page $13$, on the section about functions and continuity.