Left- and right-sided continuity

In mathematics, left- and right-sided continuity describes the property that a function is only continuous when viewed from one side. By “dividing” the continuity into left-hand and right-hand continuity, one has the property of a continuous function of “not making jumps”, divided into the properties of not making jumps when approaching the point under consideration from the left or right .

Mathematically, one-sided continuity is described using one-sided limit values. A one-sided limit value only approaches the value from one side, so a distinction is made between a left-hand and right-hand limit.

Graph of a function that is continuous on the left . ${\displaystyle x_{0}}$ ${\displaystyle f}$

definition

A function is called continuous on the left in one point of its domain if the equation for the limit on the left ${\displaystyle f}$ ${\displaystyle x_{0}\in D_{f}\subseteq \mathbb {R} }$

${\displaystyle \lim _{x\to x_{0}-}f(x)=f(x_{0})}$

holds, equivalent to this if the restriction from to is continuous in , or also equivalent to this if the condition ${\displaystyle f}$ ${\displaystyle ]-\infty ,x_{0}]\cap D_{f}}$ ${\displaystyle x_{0}}$

${\displaystyle (a_{n})_{n\in \mathbb {N} }\to x_{0}\Longrightarrow (f(a_{n}))_{n\in \mathbb {N} }\to f(x_{0})}$

holds for all strictly monotonically increasing sequences in . ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle D_{f}}$

The concept of right-hand continuity (e.g. via strictly monotonically falling sequences) is defined analogously. The continuity of in is then equivalent to the fact that the function is continuous both on the left and on the right in . This enables points of discontinuity to be classified . ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$

Examples

The Heaviside function is continuous in 0 on the right but not on the left. In contrast, the sign function in 0 is neither continuous on the left nor on the right.

literature

• H. Heuser: Textbook of Analysis. Part 1. Vieweg + Teubner, Wiesbaden 1980, ISBN 3-519-02221-4 (17th updated edition, ibid. 2009, ISBN 978-3-8348-0777-9 ).