Suppose $D$ is a set of real numbers,  $f : D \rightarrow R$.

Definition: We say that $f$ is continuous on $D$ if and only if for any $a \in D$ , $f$ is continuous at $a$, that is, for any $a \in D$,  $$ \lim_{x \rightarrow a} f(x) = f(a) $$ or that the following condition is satisfied:
For any $a \in D $ and for any $\epsilon \gt 0$, there exists a positive number $\delta (\  \gt 0)$ where if $x \in D$ and $ |x - a| < \delta$ then $|f(x) - f(a)| < \epsilon$.

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We can break the analytic definition for continuity down into three steps needed to verify it being correctly applied.

0. "For any $a \in D $":  First consider any number, $a$, that is arbitrarily chosen in the domain (possibly restricted to) $D$.
1. "and for any $ \epsilon \gt 0$": Next consider a positive real number, $\epsilon$, that is arbitrarily chosen and usually considered to be small, that is "close to $0$".
   The number $\epsilon$ will be a control on the values of the function $f$, ensuring that these values are "close to $f(a)$".
   In fact, we want $|f(x) - f(a)| < \epsilon$; that is, we want the values of $f$, $f(x)$, to be within the interval $(f(a)- \epsilon, f(a) + \epsilon)$.
   
2. "there exists a positive number $\delta (\  \gt 0)$": Now based on the chosen $\epsilon$, the task is to find a second  positive number, $\delta$. The number $\delta$ provides a control on the number $x$ in the domain of $f$ to keep $x$ "close to $a$".
   In fact the constraint $|x - a| < \delta$ amounts to having $x$ in the interval $(a-\delta,a+\delta)$. 
3. "if $x \in D$ and $ |x - a| < \delta$ then $|f(x) - f(a)| < \epsilon$": The number $\delta$ works in the definition for the given $\epsilon$ only if when $x$ is a number with $x \in D$  and  $  |x - a| < \delta$ then it can be demonstrated that $|f(x) - f(a)| < \epsilon$. The demonstration will connect the constraint on $x$ with the actual function's definition and an analysis that eventually results in a conclusion connecting to the inequality $|f(x) - f(a)| < \epsilon$.

Comments On the Domain, $D$:

• Most commonly the domain, $D$, under consideration is an interval or a union of intervals.
• When $D = (a,b)$, the limit considered in this definition is of the same general type as in the previous definition of limit section.
• When $D = [a,b]$, the limit considered for $(a,b)$ is of the same general type as in the previous definition of limit section, while the limits considered for $a$ and $b$ are of the special one sided type either from above for $a$ or from below for $b$.

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Examples of Continuity on Domains:

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