\begin{definition}[Definition:Continuous Real Function/Half Open Interval]
Let $f$ be a real function defined on a half open interval $\hointl a b$.
Then $f$ is '''continuous on $\hointl a b$''' {{iff}} it is:
:$(1): \quad$ continuous at every point of $\openint a b$
:$(2): \quad$ continuous on the left at $b$.
Let $f$ be a real function defined on a half open interval $\hointr a b$.
Then $f$ is '''continuous on $\hointr a b$''' {{iff}} it is:
:$(1): \quad$ continuous at every point of $\openint a b$
:$(2): \quad$ continuous on the right at $a$.
\end{definition}