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C04 · Unit C1 Limits and continuity

Continuity and repairing piecewise functions

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 1.10–1.13
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Continuity at a requires f(a) to exist, a two-sided limit to exist, and equality of the two. At an endpoint use the relevant one-sided limit. Polynomials are continuous everywhere; rational functions are continuous where their denominator is nonzero. A removable discontinuity can be repaired by one value; a jump with unequal side limits cannot.

Worked example

Let f(x) equals kx+1 for x<2 and x squared minus 1 for x is greater than or equal to 2. The right value and limit are 3. Left limit is 2k+1, so continuity requires k equals 1. This aligns the pieces at the join without proving they have the same slope.

Explore the mathematics

Continuity and repairing piecewise functions

Approach a missing point from both sides without substituting into an undefined expression.

Setting 3 of 5

Scroll sideways to see the full diagram.

What value do the outputs approach near x=2?xy1234f(x)=(x²−4)/(x−2), x≠2; h=0.125

Predict before continuing

What value do the outputs approach near x=2?

Read the complete explanation
  1. What value do the outputs approach near x=2?
  2. Approach from both sides; the point at x=2 is missing.
  3. At x=2±0.125, the outputs are 4±0.125.
  4. The limit is 4 although f(2) is undefined. Defining f(2)=4 repairs this removable discontinuity.
See the idea: Continuity and repairing piecewise functions
See the idea: Continuity and repairing piecewise functionsWith k=1, the left-hand line x+1 meets the right-hand curve x²−1 at (2,3). Their values agree, giving continuity, but their slopes 1 and 4 differ.xy001428312416join (2,3)

With k=1, the left-hand line x+1 meets the right-hand curve x²−1 at (2,3). Their values agree, giving continuity, but their slopes 1 and 4 differ.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Continuity does not guarantee differentiability. Matching function values is only the first joining condition.

Investigate and explain

Create a removable discontinuity, a jump and a corner. Label which can be repaired by changing one function value.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.