Continuous Extension to a Point

Definition

A continuous extension is a method of “repairing” a removable discontinuity by defining or redefining a function’s value at a point $c$ to match the limit of the function as $x \to c$.

Why It Matters

It allows us to ‘repair’ mathematical models that have tiny, insignificant gaps, making them useful for global analysis.

Core Concepts

• Condition for Extension: $\lim_{x \to c} f(x) = L$ must exist and be finite.
  • How to read: “The limit as x approaches c of f of x equals L.”
  • Meaning: A finite limit must exist—the discontinuity is removable, not a jump or asymptote.
• The Extended Function: $F(x) = \begin{cases} f(x), & x \neq c \\ L, & x = c \end{cases}$
  • How to read: “The function F of x equals f of x when x is not equal to c, and equals L when x equals c.”
  • Meaning / when to use: Plug the hole by defining $F(c) = L$; the new function is continuous at $c$.
• Outcome: The new function $F(x)$ is continuous at $c$ because its value at $c$ now equals its limit at $c$.

Connected Concepts

• Limits
• Continuity at a Point
• Types of Discontinuity
• The Limit of sin(theta)/theta