Exploring Types of Discontinuities

 A function is continuous at \( x = a \) if all three conditions hold: 1. \( f(a) \) is defined. 2. \( \lim_{x \to a} f(x) \) exists. 3. \( \lim_{x \to a} f(x) = f(a) \). If any of these fail, the function is discontinuous at $x = a$.

Main Types of Discontinuities:  

1. Removable Discontinuity

• Occurs when \( \lim_{x \to a} f(x) \) exists, but \( f(a) \) is either:
  • Not defined, or
  • Defined but not equal to the limit.
• On the graph: A “hole” at \( x = a \).
• Reason: Often caused by a common factor canceling out in rational functions.

Example: \( f(x) = \dfrac{x^2 – 1}{x – 1}, \; x \neq 1 \) Here: \( x^2 – 1 = (x – 1)(x + 1), \text{ so } f(x) = x + 1 \text{ for } x \neq 1. \) Thus: \( \lim_{x \to 1} f(x) = 2, \text{ but } f(1) \text{ undefined.} \)

2. Jump Discontinuity

• Occurs when both one-sided limits exist but are not equal: \( \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x). \)
• On the graph: A jump (a gap between two points).
• Reason: Typically appears in piecewise functions.

Example: \( f(x) = \begin{cases} 2x + 1, & x < 0 \\ x – 3, & x \ge 0 \end{cases} \) Here: \( \lim_{x \to 0^-} f(x) = 1, \quad \lim_{x \to 0^+} f(x) = -3. \) Different values → jump discontinuity at \( x = 0 \).

3. Infinite Discontinuity

• Occurs when function values become unbounded near \( x = a \): \( \lim_{x \to a^-} f(x) = \pm \infty \text{ or } \lim_{x \to a^+} f(x) = \pm \infty. \)
• On the graph: A vertical asymptote.
• Reason: Usually caused by a zero in the denominator of a rational function.

Example: \( f(x) = \dfrac{1}{x – 2}. \) As \( x \to 2 \), \( f(x) \to \pm \infty \text{ (vertical asymptote at } x = 2\text{).} \)