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