A function $f$ is continuous at $x\=a$ is small changes in $x$ in the vicinity of $a$ result in small changes in the values of $f$. In other words, small changes imply small changes. This is the notion that the formal definition below captures in mathematical language.

Definition 1.6.1 formalizes the notion that near $x\=a$, the values of $f\left(x\right)$ stray only a little from $f\left(a\right)$ as long as the values of $x$ stray only a little from $a$. In other words, small changes in $x$ induce only small changes in $f$.

Implicit in Definition 1.6.1 is the existence of the number $f\left(a\right)$. In other words, $f$ must be defined at $x\=a$. Moreover, Definition 1.6.1 has more than a passing similarity to Definition 1.2.1, the formal definition of a limit. Indeed, Definition 1.6.1 could be rephrased as "the limit at $x\=a$ is the function value at $x\=a$." In other words, if $f$ is defined at $x\=a$, and $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$, then $f$ is continuous at $x\=a$. Consequently, the test for continuity amounts to checking for the three items in Table 1.6.1.

Properly understood, (3) in Table 1.6.1 is all that needs to be remembered. If it is assumed that all the symbols in $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$ are defined (that is, the quantities exist), then this equality captures the essence of Definition 1.6.1.

If $x\=a$ is the endpoint of a closed interval that is the domain of a function $f$, the question of continuity at $a$ is resolved by considering the appropriate one-sided limit at $x\=a$. Thus, if $a$ is a left endpoint, then $f$ is continuous at $x\=a$ if $\underset{x\to a\+}{lim}f\left(x\right)\=f\left(a\right)$, whereas if $a$ is a right endpoint, then $f$ is continuous at $x\=a$ if $\underset{x\to a-}{lim}f\left(x\right)\=f\left(a\right)$.

A function that is not continuous at a point is said to be discontinuous at that point. A function can be discontinuous at a point in one of the four ways listed in Table 1.6.2.

Continuity is a "point property" that is extended to an interval if every point in the interval has that property.

It is impossible to test for continuity at every point in an interval. Hence, it is useful to know how primitive continuous functions combine to form (or not form) other continuous functions. To this end, see Table 1.6.3. However, before scanning the table, it is useful to sharpen intuition with a few observations:

This second observation needs some explaining. For example, if $f\left(x\right)\=x\+1\/x$ and $g\left(x\right)\=x^2-1\/x$, then both functions are discontinuous at $x\=0$, but the sum, $x\+x^2$, is the rule for a function that is continuous at $x\=0$. Looked at more critically, note that the domains of both $f$ and $g$ do not contain $x\=0$, so the domain of their sum should not contain it either (a function is a rule and a domain), and the sum is not continuous at $x\=0$. But under the "convention" that the domain of a rule is the largest set for which it is defined, the domain of $x\+x^2$ would be all the reals, and the sum would be continuous at $x\=0$.

It would seem that the contents of Table 1.6.3 shouldn't be taken lightly.

The key ideas in this section are summarized in Table 1.6.4