Right Continuous Function
A Right Continuous Function is a function, [math]\displaystyle{ f }[/math], with point [math]\displaystyle{ c }[/math] such that there is no jump when [math]\displaystyle{ f }[/math]'s limit point is approached from the right.
- Context:
- For any number [math]\displaystyle{ ε \gt 0 }[/math] there exists some number [math]\displaystyle{ \delta \gt 0 }[/math] such that for all [math]\displaystyle{ x }[/math] in the domain with [math]\displaystyle{ c \lt x \lt c + \delta }[/math], the value of [math]\displaystyle{ f(x) }[/math] will satisfy [math]\displaystyle{ |f(x) - f(c)| \lt \varepsilon . }[/math]
- Example(s):
- Counter-Example(s):
- See: a Bivariate CDF, Sample Path.
References
2011
- (Wikipedai, 2011) ⇒ http://en.wikipedia.org/wiki/Continuous_function#Directional_and_semi-continuity
- QUOTE: Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. More formally, ƒ is said to be right-continuous at the point c if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c < x < c + δ, the value of ƒ(x) will satisfy :[math]\displaystyle{ |f(x) - f(c)| \lt \varepsilon.\, }[/math]
This is the same condition as for continuous functions, except that it is required to hold for x strictly larger than c only. Requiring it instead for all x with c − δ < x < c yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.
- QUOTE: Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. More formally, ƒ is said to be right-continuous at the point c if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c < x < c + δ, the value of ƒ(x) will satisfy :[math]\displaystyle{ |f(x) - f(c)| \lt \varepsilon.\, }[/math]
2009
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Continuous_function#Directional_continuity
- A function may happen to be continuous in only one direction, either from the "left" or from the "right". A right-continuous function is a function which is continuous at all points when approached from the right. Technically, the formal definition is similar to the definition above for a continuous function but modified as follows:
The function ƒ is said to be right-continuous at the point c if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c < x < c + δ, the value of ƒ(x) will satisfy |f(x) - f(c)| < \varepsilon.\,
Notice that x must be larger than c, that is on the right of c. If x were also allowed to take values less than c, this would be the definition of continuity. This restriction makes it possible for the function to have a discontinuity at c, but still be right continuous at c, as pictured.
Likewise a left-continuous function is a function which is continuous at all points when approached from the left, that is, c − δ < x < c.
- A function may happen to be continuous in only one direction, either from the "left" or from the "right". A right-continuous function is a function which is continuous at all points when approached from the right. Technically, the formal definition is similar to the definition above for a continuous function but modified as follows:
2007
- (Trivedi & Zimmer, 2007) ⇒ Pravin K. Trivedi, and David M. Zimmer. (2007). “Copula modeling: an introduction for practitioners.” In: Volume 1 of Foundations and Trends in Econometrics. Now Publishers Inc. ISBN:1601980205
- QUOTE: The following conditions are necessary and sufficient for a right-continuous function to be bivariate cdf: ...