PlanetMath: one-sided continuity

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one-sided continuity

(Definition)

The real function is continuous from the left in the point iff

$\displaystyle \lim_{x\to x_0-}f(x) = f(x_0).$

The real function is continuous from the right in the point iff

$\displaystyle \lim_{x\to x_0+}f(x) = f(x_0).$

The real function 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 and from the left continuous at .

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.

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"one-sided continuity" is owned by pahio.

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Also defines:

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

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Cross-references: mantissa function, integer, ceiling function, open interval, continuous at, closed interval, continuous, iff, point, real function
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This is version 3 of one-sided continuity, born on 2008-04-02, modified 2008-04-03.
Object id is 10469, canonical name is OneSidedContinuity.
Accessed 1121 times total.

Classification:

AMS MSC:

26A06 (Real functions :: Functions of one variable :: One-variable calculus)

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