# 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

### 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: ...