Continuity from Left, from Right, and over an Interval

The single-point test of §1 extends naturally to one-sided continuity and to continuity across a whole stretch of the domain.

One-sided continuity. A function $f$ is left-continuous at $x = a$ if $\displaystyle\lim_{x \to a^-} f(x) = f(a)$, and right-continuous at $x = a$ if $\displaystyle\lim_{x \to a^+} f(x) = f(a)$. A function is continuous at $a$ (in the full two-sided sense of §1) precisely when it is both left-continuous and right-continuous there. One-sided continuity matters most at the endpoints of an interval, where the function is only approached from one side.

Continuity over an open interval. A function $f$ is continuous on an open interval $(a, b)$ if it is continuous at every point of that interval. Since an open interval contains none of its endpoints, only the ordinary two-sided test is used at each interior point.

Continuity over a closed interval. A function $f$ is continuous on a closed interval $[a, b]$ if:

• it is continuous at every interior point $x$ with $a < x < b$ (two-sided test), and
• it is right-continuous at the left endpoint $a$, i.e. $\displaystyle\lim_{x \to a^+} f(x) = f(a)$, and
• it is left-continuous at the right endpoint $b$, i.e. $\displaystyle\lim_{x \to b^-} f(x) = f(b)$.

At an endpoint only the inside one-sided limit can be taken, because the function is only defined on one side within the interval — this is why closed-interval continuity uses one-sided tests at the two ends and the full two-sided test only in between.

Continuous function. If $f$ is continuous at every point of its domain, it is simply called a continuous function. Polynomials, for instance, are continuous functions on all of $\mathbb{R}$; a rational function is continuous at every point of its domain (that is, everywhere except where its denominator is zero, which points are excluded from the domain to begin with).

Illustration. The function $f(x) = \dfrac{1}{x-3}$ is continuous at every real number except $x = 3$, where it is not even defined (the denominator vanishes). So $f$ is a continuous function on its domain — the set of all $x \neq 3$ — and, for example, is continuous on the closed interval $[4, 10]$ (which avoids $3$ entirely) but not on $[0, 5]$ (which contains the excluded point $3$). …