Continuity and Discontinuity in Functions

Key Concepts

• A continuous graph has no jumps or breaks.
• Types of discontinuity:
  • Hole (Removable Discontinuity): A point where the graph is undefined but can be defined by limit.
  • Jump Discontinuity (Non-Removable): The graph has different left-hand and right-hand limits.
  • Infinite Discontinuity (Non-Removable): The graph approaches positive or negative infinity at a point, typically associated with vertical asymptotes.

Identifying Points of Discontinuity

1. Example: 1/x²

   • Vertical asymptote at x = 0 (infinite discontinuity).

2. Example: 5/(x + 2)

   • Vertical asymptote at x = -2 (infinite discontinuity).

3. Example: (3x + 2)/(x + 2)(x - 5)

   • Vertical asymptote at x = 5 (infinite discontinuity).
   • Hole at x = -2 (removable discontinuity).

4. Example: |x|/x

   • Undefined at x = 0. Graph shows
     • For x > 0, f(x) = 1
     • For x < 0, f(x) = -1
   • Result: Jump Discontinuity (non-removable).

5. Piecewise Function Example:

   • f(x) = 5x + 3 (x < 1)
   • f(x) = x² + 4 (1 ≤ x < 2)
   • f(x) = x³ (x ≥ 2)
   • Check points:
     • At x = 1, f(1) = 8 (from first function) and f(1) = 5 (from second function) → Discontinuous.
     • At x = 2, f(2) = 8 for both functions → Continuous.

Finding Constants for Continuity

1. Example: cx + 3 (x < 2) and 3x + c (x ≥ 2)

   • Set equal at x = 2:
     • 2c + 3 = 6 + c
     • Solving gives c = 3.

2. Example: ax - 2 (x < 3) and x² - 5 (x ≥ 3)

   • Set equal at x = 3:
     • 3a - 2 = 9 - 5
     • Solving gives a = 2.

3. Finding a and b for piecewise function:

   • Condition: ax + 5 (x < 1) and x² - bx + 9 (1 ≤ x < 4)
   • Set equal at x = 1:
     • a + 5 = 10 - b → a + b = 5.
   • Set equal at x = 4:
     • 16 - 4b + 9 = 16a - 7 → 25 = 16a - 7
     • Solving gives a = 2 and substituting back gives b = 3.