What Makes Piecewise Functions Different?
A piecewise-defined function uses different formulas on different parts of its domain. For example, one rule might apply when , while another applies when .
The key idea is simple: differentiate each formula on the interval where it applies. But there is one special issue at the point where the pieces meet. At that breakpoint, you cannot automatically differentiate either formula. You must check whether the function is continuous there and whether the left- and right-hand derivatives agree.
A function can be differentiable on every individual piece and still fail to be differentiable at the point where the pieces join.
Step 1: Differentiate Each Piece
Suppose
For values of , differentiate the first formula normally. For values of , differentiate the second formula normally.
This gives the derivative on each open interval. However, this does not automatically tell us what happens at the breakpoint .
Why the Breakpoint Requires Special Attention
Suppose the pieces meet at . For to exist, the function must satisfy two conditions:
- Continuity: the function must actually be continuous at .
- Matching slopes: the derivative from the left and the derivative from the right must be equal.
The one-sided derivative conditions are
If either condition fails, the function is not differentiable at that point.
Worked Examples
Find for the function
The important point is that differentiating the pieces separately gives the derivative on the open intervals. We still need to examine the breakpoint itself.
Determine whether the function
is differentiable at .
Determine whether
is differentiable at .
So the function is continuous at .
Find the value of that makes
continuous and differentiable at .
The right piece must have the same value:
The constant must satisfy both the continuity and differentiability requirements. On AP Calculus questions, checking only one of these conditions is a common trap.
A Reliable AP Calculus Method
When a question asks whether a piecewise function is differentiable at a breakpoint, use this sequence every time:
- Identify the breakpoint. This is the value of where the formula changes.
- Check continuity. Compare the left-hand limit, right-hand limit, and actual function value.
- Differentiate each piece. Use the appropriate derivative rules on each side.
- Evaluate the one-sided derivatives at the breakpoint.
- Compare them. If the left and right derivatives are equal, the derivative exists.
Common Mistakes
A function can be continuous but have a corner or cusp. Continuity is necessary for differentiability, but it is not sufficient. Always compare the one-sided derivatives.
Pay close attention to inequalities such as versus . The piece that includes the breakpoint determines the actual value of the function there, while the one-sided derivatives come from the appropriate sides.
Two derivative formulas can have matching values at the breakpoint while the original function is discontinuous there. Differentiability requires continuity first.
If a function has different formulas on different intervals, its derivative will usually have different formulas on those intervals too. That's completely normal.
Practice Problems
Find for
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Determine whether the function
is differentiable at .
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Find the value of that makes
continuous and differentiable at .
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A function is defined by
Determine whether exists.
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