Piecewise Function Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Determine whether f(x)={x+1x<23x=22x−1x>2f(x) = \begin{cases} x + 1 & x < 2 \\ 3 & x = 2 \\ 2x - 1 & x > 2 \end{cases} is continuous at x=2x = 2.

Solution

  1. 1
    Check lim⁡x→2−f(x)\lim_{x \to 2^-} f(x): as x→2x \to 2 from the left, f(x)=x+1→3f(x) = x+1 \to 3.
  2. 2
    Check lim⁡x→2+f(x)\lim_{x \to 2^+} f(x): as x→2x \to 2 from the right, f(x)=2x−1→3f(x) = 2x-1 \to 3.
  3. 3
    Check f(2)=3f(2) = 3. Since lim⁡x→2−f(x)=lim⁡x→2+f(x)=f(2)=3\lim_{x\to2^-}f(x) = \lim_{x\to2^+}f(x) = f(2) = 3, the function is continuous at x=2x=2.

Answer

ff is continuous at x=2x = 2
Continuity at a point requires that both one-sided limits exist, are equal, and match the function's value at that point. Here all three conditions are satisfied even though the piecewise function uses three separate rules.

About Piecewise Function

A piecewise function is defined by different formulas on different non-overlapping intervals of its domain, with the applicable formula determined by the input value.

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More Piecewise Function Examples

Example 1 easy

Evaluate [formula] at [formula], [formula], and [formula].

Example 3 easy

Given [formula], evaluate [formula], [formula], and [formula].

Example 4 hard

Find the value of [formula] so that [formula] is continuous at [formula].

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