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Continuity of a Piecewise Function (Step-by-Step Example with Solution)

  • Writer: Tyler Buffone
    Tyler Buffone
  • Feb 1
  • 1 min read

Updated: Oct 17

Abstract landscape artwork featuring smooth, flowing curves that blend seamlessly in shades of blue, white, and gold, symbolizing continuity and smooth transitions.

Example


Let a be a real constant. Define:


Math equation on a dark background: F(x) = 1-x² if x<3; ax²+1 if x≥3. White handwritten text and symbols convey a formal mood.

Find the value of a that makes f(x) continuous at x = 3. Explain your reasoning.


Solution


The function f(x) is a piecewise function consisting of two polynomials, and the transition occurs at the point x = 3.


Each piece is individually continuous (since polynomials are continuous everywhere), so f(x) is automatically continuous on the real number line except possibly at x = 3.


To ensure that f(x) is continuous at x = 3, we must have:


Math equation on dark background: lim as x approaches 3 of F(x) equals F(3) equals a(3)² + 1 equals 9a + 1, with the number 3 in blue.

For this to be true, the left-hand and right-hand limits must both equal 9a + 1.


We have that:


Math equations on a dark background showing limits as x approaches 3 plus and minus. Text includes calculations and equals negative 8.

So, we ultimately need that:


9a + 1 = -8


Solving for a, we get that:


a = -1


Therefore, the value of a which will guarantee that f(x) is continuous at x = 3 is - 1.



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