Definition:Continuity

= Real Function =

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 a 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 a 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 $$a$$

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

Then $$f$$ is continuous on $$\left({a \, . \, . \, b}\right)$$ iff it is continuous at every point of $$\left({a \, . \, . \, b}\right)$$.

Closed Interval
Let $$f$$ be a real function defined on a closed interval $$\left[{a \,. \, . \, b}\right]$$.

Then $$f$$ is ''continuous on $$\left[{a \,. \, . \, b}\right]$$'' iff it is:
 * continuous at every point of $$\left({a \, . \, . \, b}\right)$$;
 * continuous on the left at $$b$$;
 * continuous on the right at $$a$$.

That is, if $$f$$ is to be continuous over the whole of a closed interval, it needs to be continuous at the end points as well. However, because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end point that the function is defined.

Half Open Intervals
Similar definitions apply to half open intervals.

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

Then $$f$$ is continuous on $$\left({a \, . \, . \, b}\right]$$ iff it is:
 * continuous at every point of $$\left({a \, . \, . \, b}\right)$$;
 * continuous on the left at $$b$$.

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

Then $$f$$ is ''continuous on $$\left[{a \,. \, . \, b}\right)$$'' iff it is:
 * continuous at every point of $$\left({a \, . \, . \, b}\right)$$;
 * continuous on the right at $$a$$.