Types of Continuity and Discontinuity Math Example 4

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Example 4

medium
Find the value of cc that makes h(x)={cx+1x≤2x2−1x>2h(x) = \begin{cases} cx + 1 & x \leq 2 \\ x^2 - 1 & x > 2 \end{cases} continuous at x=2x = 2.

Solution

  1. 1
    For continuity: left limit = right limit = function value.
  2. 2
    Right limit: lim⁡x→2+(x2−1)=3\lim_{x\to 2^+}(x^2-1) = 3.
  3. 3
    Left limit and value: lim⁡x→2−(cx+1)=2c+1\lim_{x\to2^-}(cx+1) = 2c+1.
  4. 4
    Set equal: 2c+1=3⇒c=12c+1 = 3 \Rightarrow c = 1.

Answer

c=1c = 1
At a boundary point, continuity requires both one-sided limits to match. Set the two expressions equal at x=2x=2 and solve for cc.

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 ±∞.

Learn more about Types of Continuity and Discontinuity →

More Types of Continuity and Discontinuity Examples

Example 1 easy

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

Example 2 medium

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

Example 3 easy

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

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