Definition:Continuous Real Function/Half Open Interval

Definition
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:
 * $(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
 * $(2): \quad$ 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:
 * $(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$
 * $(2): \quad$ continuous on the right at $a$.