one-sided continuity

The real function $f$ is continuous from the left in the point  $x=x_0$  iff

$$\underset{x\to x_0-}{lim}f(x)=f(x_0).$$

The real function $f$ is continuous from the right in the point  $x=x_0$  iff

$$\underset{x\to x_0+}{lim}f(x)=f(x_0).$$

The real function $f$ is continuous on the closed interval  $[a,b]$  iff it is continuous at all points of the open interval  $(a,b)$,  from the right continuous at $a$ and from the left continuous at $b$.

Examples.  The ceiling function $\lceil x\rceil$ is from the left continuous at each integer, the mantissa function $x-\lfloor x\rfloor$ is from the right continuous at each integer.