Left Continuous Function
A Left Continuous Function is a Continuous Function that ...
- (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 is continuous if and only if it is both right-continuous and left-continuous.