Continuity Over an Interval

Continuity has so far been discussed only at a single point. The same idea extends naturally to a whole interval.

Let $(a,b)$ be an open interval. If $f$ is continuous at every point $x\in(a,b)$, then $f$ is said to be continuous on $(a,b)$.

Now consider $f$ defined on the half-open interval $[a,b)$. If $f$ is continuous on $(a,b)$ and continuous to the right of $a$ — that is, $\displaystyle\lim_{x\to a^+} f(x)=f(a)$ — then $f$ is continuous on $[a,b)$. Similarly, consider $f$ defined on $(a,b]$: if $f$ is continuous on $(a,b)$ and continuous to the left of $b$ — that is, $\displaystyle\lim_{x\to b^-} f(x)=f(b)$ — then $f$ is continuous on $(a,b]$. …