# one-sided continuity

The real function $f$ is continuous from the left in the point  $x=x_{0}$  iff

 $\lim_{x\to x_{0}-}f(x)=f(x_{0}).$

The real function $f$ is continuous from the right in the point  $x=x_{0}$  iff

 $\lim_{x\to x_{0}+}f(x)=f(x_{0}).$

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.

 Title one-sided continuity Canonical name OnesidedContinuity Date of creation 2013-03-22 17:57:50 Last modified on 2013-03-22 17:57:50 Owner pahio (2872) Last modified by pahio (2872) Numerical id 6 Author pahio (2872) Entry type Definition Classification msc 26A06 Related topic OneSidedLimit Related topic OneSidedDerivatives Related topic OneSidedContinuityBySeries Defines continuous from the left Defines continuous from the right Defines from the left continuous Defines from the right continuous Defines continuous on closed interval