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.

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