Definition:Continuity

Continuity at a Point
Let $$f$$ be a real function defined on an open interval $$\left({a \, . \, . \, b}\right)$$.

Let $$c \in \left({a \, . \, . \, b}\right)$$.

Then $$f$$ is continuous at (the point) $$c$$ iff $$\lim_{x \to c} f \left({x}\right) = f \left({c}\right)$$.

Loosely speaking, this means that continuity at a point is defined as the graph of $$f$$ not having a "break" at $$c$$.

Note that from the definition of a limit of a function, it is necessary that $$f \left({c}\right)$$ is both:
 * the limit from the left, and
 * the limit from the right

of $$f \left({x}\right)$$ as $$x$$ tends to $$c$$ from both above $$c$$ and below $$c$$.

Continuous on the Left
Let $$f$$ be a real function defined on an half open interval $$\left({a \, . \, . \, b}\right]$$.

Let $$\lim_{x \to b^-} f \left({x}\right) = f \left({b}\right)$$.

Then $$f$$ is continuous on the left at the point $$b$$

Continuous on the Right
Let $$f$$ be a real function defined on an half open interval $$\left[{a \,. \, . \, b}\right)$$.

Let $$\lim_{x \to a^+} f \left({x}\right) = f \left({a}\right)$$.

Then $$f$$ is continuous on the right at the point $$a$$

Continuity on an Arbitrary Domain
Let $$f$$ be a real function on a domain $$D \subseteq \mathbb{R}$$.

Then $$f$$ is continuous on $$D$$ iff the limit $$\lim_{x \to c} f \left({x}\right) = f \left({c}\right)$$ for all $$c \in D$$.

If the domain of the function is a half open real interval or a closed real interval, then obviously there are not both right hand and left hand limits for $$f$$ at the endpoints.

Therefore, the function is defined as being:
 * continuous at the left endpoint of the interval $$\left[{a \, . \, . \, b}\right]$$ or $$\left[{a \, . \, . \, b}\right)$$ if $$f \left({a}\right) = \lim_{x \to a^+} f(x)$$, and
 * continuous at the right endpoint of the interval $$\left[{a \, . \, . \, b}\right]$$ or $$\left({a \, . \, . \, b}\right]$$ if $$f \left({b}\right) = \lim_{x \to b^-} f(x)$$.