Continuity of a Function: Three-Step Test

Three-Step Continuity Test

1. Function Definition

   • Function must be defined at a specific point a.
   • ( f(a) ) must exist.

2. Limit Existence

   • Show that ( \lim_{{x \to a}} f(x) ) exists.
   • Prove ( \lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x) ).

3. Equality of Limit and Function Value

   • Confirm ( \lim_{{x \to a}} f(x) = f(a) ).

Example Applications

Example 1: Function ( f(x) = \begin{cases} \sqrt{x + 2}, & x < 2 \ x^2 - 2, & 2 \leq x < 3 \ 2x + 5, & x \geq 3 \end{cases} )

• Check Continuity at ( x = 2 ):

  1. Function Defined: ( f(2) = 2^2 - 2 = 2 )
  2. Limit Existence:
     • ( \lim_{{x \to 2^-}} f(x) = \sqrt{2 + 2} = 2 )
     • ( \lim_{{x \to 2^+}} f(x) = 2^2 - 2 = 2 )
     • Limits are equal; limit exists.
  3. Equality of Limit and Function Value:
     • ( \lim_{{x \to 2}} f(x) = 2 = f(2) )
  • Conclusion: Continuous at ( x = 2 ).

• Check Continuity at ( x = 3 ):

  1. Function Defined: ( f(3) = 2 \times 3 + 5 = 11 )
  2. Limit Existence:
     • ( \lim_{{x \to 3^-}} f(x) = 3^2 - 2 = 7 )
     • ( \lim_{{x \to 3^+}} f(x) = 2 \times 3 + 5 = 11 )
     • Limits are not equal; limit does not exist.
  • Conclusion: Discontinuous at ( x = 3 ) (Jump Discontinuity)._

Example 2: Function ( f(x) = \begin{cases} 2x + 5, & x < 1 \ x^2 + 2, & x > 1 \ 5, & x = 1 \end{cases} )

• Check Continuity at ( x = 1 ):
  1. Function Defined: ( f(1) = 5 )
  2. Limit Existence:
     • ( \lim_{{x \to 1^-}} f(x) = 2 \times (-1) + 5 = 3 )
     • ( \lim_{{x \to 1^+}} f(x) = 1^2 + 2 = 3 )
     • Limits are equal; limit exists.
  3. Equality of Limit and Function Value:
     • ( \lim_{{x \to 1}} f(x) = 3 \neq f(1) = 5 )
  • Conclusion: Discontinuous at ( x = 1 ) (Removable Discontinuity)._

Types of Discontinuities

• Jump Discontinuity:

  • Occurs when left-hand and right-hand limits are unequal.
  • Non-removable discontinuity.

• Removable Discontinuity:

  • Occurs when ( \lim_{{x \to a}} f(x) ) exists but is not equal to ( f(a) ).
  • Appears as a hole in the graph.

• Infinite Discontinuity:

  • Occurs when limits approach infinity.
  • Not applicable unless function involves infinity._