Types of Continuity and Discontinuity Math Example 1

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Example 1

easy
Classify the discontinuity of f(x)=x2โˆ’4xโˆ’2f(x) = \dfrac{x^2 - 4}{x - 2} at x=2x = 2.

Solution

  1. 1
    At x=2x=2: denominator =0= 0, so f(2)f(2) is undefined.
  2. 2
    Simplify (for xโ‰ 2x \neq 2): (xโˆ’2)(x+2)xโˆ’2=x+2\frac{(x-2)(x+2)}{x-2} = x+2.
  3. 3
    Limit: limโกxโ†’2(x+2)=4\lim_{x\to 2}(x+2) = 4. The limit exists.
  4. 4
    Since the limit exists but f(2)f(2) is undefined, this is a removable discontinuity (hole at (2,4)(2,4)).

Answer

Removable discontinuity at x=2x = 2 (hole at the point (2,4)(2, 4)).
A removable discontinuity occurs when the two-sided limit exists but doesn't equal the function value (or the function is undefined there). It can be 'removed' by defining f(2)=4f(2) = 4.

About Types of Continuity and Discontinuity

Continuity types classify how a function can fail to be continuous at a point. A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(a). A jump discontinuity occurs when left and right limits differ. An infinite discontinuity occurs when the function approaches ยฑโˆž.

Learn more about Types of Continuity and Discontinuity โ†’

More Types of Continuity and Discontinuity Examples

Example 2 medium

Determine whether the piecewise function [formula] is continuous at [formula].

Example 3 easy

Classify the discontinuity of [formula] at [formula].

Example 4 medium

Find the value of [formula] that makes [formula] continuous at [formula].

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