Continuity from the Right and from the Left

When a function switches formula at a point, continuity there is checked one side at a time.

A function $f(x)$ is said to be continuous from the right at $x=a$ if $\displaystyle\lim_{x\to a^+} f(x)=f(a)$, and continuous from the left at $x=a$ if $\displaystyle\lim_{x\to a^-} f(x)=f(a)$. If $f$ is continuous from both the right and the left at $a$, then it is continuous at $a$, because then

Illustration 3. Consider $f(x)=\lfloor x\rfloor$ (the greatest-integer / floor function) on the interval $[2,4)$, so that $f(x)=2$ for $x\in[2,3)$ and $f(x)=3$ for $x\in[3,4)$ (Fig. 8.5). To test continuity at $x=3$: here $f(3)=3$, and

Since $\displaystyle\lim_{x\to3^-} f(x)\ne\lim_{x\to3^+} f(x)$, the function $f(x)=\lfloor x\rfloor$ is discontinuous at $x=3$ — the left and right approaches settle at different heights, so there is a visible step in Fig. 8.5.

Illustration 4. Consider $f(x)=x^2+\tfrac32$ for $0\le x\le3$, and $f(x)=5x-4.5$ for $3<x\le5$ (Fig. 8.6). At $x=3$, $f(3)=3^2+\tfrac32=10.5$, and …