Definition:Continuity

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