# Right Continuous Function

(Redirected from Right-Continuous Function)
Jump to: navigation, search

A Right Continuous Function is a function, $f$, with point $c$ such that there is no jump when $f$'s limit point is approached from the right.

• Context:
• For any number $ε \gt 0$ there exists some number $\delta \gt 0$ such that for all $x$ in the domain with $c \lt x \lt c + \delta$, the value of $f(x)$ will satisfy $|f(x) - f(c)| \lt \varepsilon .$
• Example(s):
• Counter-Example(s):
• See: a Bivariate CDF, Sample Path.

## References

### 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.