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# 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.

Defines:

continuous from the left, continuous from the right, from the left continuous, from the right continuous, continuous on closed interval

Related:

OneSidedLimit, OneSidedDerivatives, OneSidedContinuityBySeries

Type of Math Object:

Definition

Major Section:

Reference

Parent:

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## Mathematics Subject Classification

26A06*no label found*

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