Continuous from the right

Definition

A function $f(x)$ is continuous from the right at $x = c$ if $\lim_{{x \to c^+}} f(x) = f(c)$. This means that as $x$ approaches $c$ from values greater than $c$, the function value approaches $f(c)$.

5 Must Know Facts For Your Next Test

1. For a function to be continuous from the right at $x = c$, $\lim_{{x \to c^+}} f(x)$ must exist and equal $f(c)$.
2. Continuity from the right does not imply general continuity; left-hand limits must also be considered for overall continuity.
3. Graphically, being continuous from the right means there is no jump or hole in the graph when approaching from the right side of point $c$.
4. To check for right-hand continuity, evaluate both $\lim_{{x \to c^+}} f(x)$ and the function value at that point, $f(c)$.
5. Right-hand continuity is used in piecewise functions to ensure smooth transitions between pieces.

Review Questions

• What does it mean for a function to be continuous from the right at a point?
• How do you mathematically verify if a function is continuous from the right at $x = c$?
• Why is it important to consider both left-hand and right-hand limits when discussing overall continuity?

Related terms

Left-Hand Limit: $\lim_{{x \to c^-}} f(x)$ is the value that $f(x)$ approaches as $x$ approaches $c$ from values less than $c$.

Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

Discontinuity: A point where a function is not continuous, which can occur due to jumps, holes, or vertical asymptotes.